ON q-INTEGRALS OVER ORDER POLYTOPES
JANG SOO KIM AND DENNIS STANTON
Abstract. A combinatorial study of multiple q-integrals is developed. This includes a q-volume of a convex polytope, which depends upon the order of q-integration. A multiple q-integral over an order polytope of a poset is interpreted as a generating function of linear extensions of the poset. Specific modifications of posets are shown to give predictable changes in q-integrals over their respective order polytopes. This method is used to combinatorially evaluate some generalized q-beta integrals. One such application is a combinatorial interpretation of a q-Selberg integral. New generating functions for generalized Gelfand-Tsetlin patterns and reverse plane partitions are established. A q-analogue to a well known result in Ehrhart theory is generalized using q-volumes and q-Ehrhart polynomials.
Contents 1. Introduction 1 2. Definitions 3 3. Properties of q-integrals 6 4. q-integrals over order polytopes 9 5. Operations on posets 11 6. Examples of q-integrals 15 6.1. The q-beta integral 15 6.2. A q-analog of Dirichlet integral 15 6.3. The general q-beta integral of Andrews and Askey 16 6.4. q-integrals of monomials over the order polytope of a forest poset 17 7. q-Selberg integrals 18 8. Reverse plane partitions 22 8.1. Shifted reverse plane partitions with fixed diagonal entries 25 8.2. Generalized Gelfand-Tsetlin patterns 26 8.3. The trace-generating function for reverse plane partitions 27 8.4. The trace-generating function for shifted reverse plane partitions 28 8.5. Reverse plane partitions and the q-Selberg integral 29 9. q-Ehrhart polynomials 31 Acknowledgements 33 References 33
1. Introduction The main object in this paper is the q-integral Z 1 n X i i f(x)dqx = (1 − q) f(q )q , 0 i=0
Date: August 5, 2016. 2010 Mathematics Subject Classification. Primary: 05A30; Secondary: 05A15, 06A07. Key words and phrases. q-integral, order polytope, q-Selberg integral, reverse plane partition, q-Ehrhart polynomial. The first author was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2061006). The second author was supported by NSF grant DMS-1148634. 1 2 JANG SOO KIM AND DENNIS STANTON which was introduced by Thomae [23] and Jackson [12]. The q-integral is a q-analogue of the R 1 n Riemann integral. Fermat used it to evaluate 0 x dx. See [2, §10.1] for more details of the history of q-integrals. Many important integrals have q-analogues in terms of q-integrals, such as q-beta integrals and q-Selberg integrals. In this paper we develop combinatorial methods to study q-integrals. The original motivation of this paper was to generalize Stanley’s combinatorial interpretation of the Selberg integral [19]
Z 1 Z 1 n Y α 1 β 1 Y 2γ Sn(α, β, γ) = ··· xi − (1 − xi) − |xi − xj| dx1 ··· dxn 0 0 i=1 1 i
2. Definitions In this section we give the necessary definitions with examples for q-integration, multiple q- integration, and order polytopes of posets. Throughout this paper we assume 0 < q < 1. We will use the following notation for q-series: n 1 − q n [n]q! [n]q = , [n]q! = [1]q[2]q ··· [n]q, = , 1 − q k q [k]q![n − k]q! n 1 (a; q)n = (1 − a)(1 − aq) ··· (1 − aq − ), (a1, a2, . . . , ak; q)n = (a1; q)n ··· (ak; q)n.
We also use the notation [n] := {1, 2, . . . , n}. We denote by Sn the set of permutations on [n].
Definition 2.1. Let π = π1π2 ··· πn ∈ Sn. An integer i ∈ [n − 1] is called a descent of π if πi > πi+1. Let Des(π) be the set of descents of π. We define des(π) and maj(π) to be the number of descents of π and the sum of descents of π, respectively. We denote by inv(π) the number of pairs (i, j) such that 1 ≤ i < j ≤ n and πi > πj. First, recall [2, §10.1] the q-integral of a function f over (a, b). Definition 2.2. For 0 < q < 1, the q-integral from a to b is defined by Z b X∞ i i i i (1) f(x)dqx = (1 − q) f(bq )bq − f(aq )aq . a i=0 In the limit as q → 1, the q-integral becomes the usual integral. It is easy to see that Z b n+1 n+1 n b − a x dqx = . a [n + 1]q We extend the definition of a q-integral to a multiple q-integral over a convex polytope. Here the ordering π of the variables is important because the iterated q-integral is not, in general, independent of the ordering.
Definition 2.3. Let π = π1 ··· πn ∈ Sn. For a function f(x) of n-variables x = (x1, . . . , xn) and a convex polytope D ∈ Rn, the q-integral of f(x) over D with respect to order π of integration is defined by
Z Z max(Dn) Z max(D1)
f(x1, . . . , xn)dqxπ1 ··· dqxπn = ··· f(x1, . . . , xn)dqxπ1 ··· dqxπn , D min(Dn) min(D1) where Di is the set of real numbers depending on the values of xπi+1 , . . . , xπn given by
Di = Di(xπi+1 , . . . , xπn ) = {yπi :(y1, . . . , yn) ∈ D, yπj = xπj for j > i}. If a convex polytope D is determined by a family Q of inequalities of the sorted variables x = (x1, . . . , xn), then we will also write Z Z f(x)dqx = f(x)dqx, where dqx = dqx1 ··· dqxn. Q D 4 JANG SOO KIM AND DENNIS STANTON
Example 2.4. We have Z Z Z d Z b f(x, y)dqxdqy = f(x, y)dqxdqy = f(x, y)dqxdqy, a x b ≤ ≤ [a,b] [c,d] c a c y d × ≤ ≤ and, for x = (x1, . . . , x6), Z Z f(x)dqx = f(x)dqx a x x x x x x b (x ,...,x ):a x x x x x x b ≤ 3≤ 1≤ 5≤ 2≤ 4≤ 6≤ { 1 6 ≤ 3≤ 1≤ 5≤ 2≤ 4≤ 6≤ } Z b Z x6 Z x6 Z x5 Z x4 Z x5 = f(x1, . . . , x6)dqx1 ··· dqx6. a a x5 a x5 x3 Note that we have Z Z lim f(x1, . . . , xn)dqxπ ··· dqxπ = f(x1, . . . , xn)dxπ ··· dxπ , q 1 1 n 1 n → D D which is independent of the ordering π if Fubini’s theorem holds. Unlike usual integrals, we do not have always Fubini’s theorem for q-integrals of well behaved functions. For example, Z 1 Z q dqx1dqx2 = , dqx2dqx1 = . 0 x x 1 1 + q 0 x x 1 1 + q ≤ 1≤ 2≤ ≤ 1≤ 2≤ We next define the q-volume of a convex polytope. Since the q-integral depends on the order of integration, we need to specify that ordering. n Definition 2.5. Suppose that D is a convex polytope in R and w = w1 ··· wn is a permutation n of the coordinates x1, x2, . . . , xn of R . Then the q-volume of D with respect to w is Z Vq(D, w) = dqw1 ··· dqwn. D
If w = x1x2 ··· xn, then we will omit w and simply write Vq(D) = Vq(D, w), that is, Z Vq(D) = dqx1 ··· dqxn. D 3 Example 2.6. If D1 = {(x1, x2, y) ∈ [0, 1] : x1 ≤ y ≤ x2}, then Z Vq(D1, yx1x2) = dqydqx1dqx2, 0 x y x 1 ≤ 1≤ ≤ 2≤ 3 and if D2 = {(x1, x2, x3) ∈ [0, 1] : x3 ≤ x1 ≤ x2}, then Z Vq(D2) = dqx1dqx2dqx3. 0 x x x 1 ≤ 3≤ 1≤ 2≤ In most of this paper we will integrate over order polytopes of partially ordered sets (posets). Definition 2.7. If P is a poset with n elements, a labeling of P is a bijection ω : P → [n]. A pair (P, ω) of a poset P and its labeling ω is called a labeled poset. If ω(x) ≤ ω(y) for any x ≤P y, we say that ω is a natural labeling of P , or P is naturally labeled. We need some rudiments of P -partition theory, which appear in [22, Chapter 3]. Definition 2.8. Let (P, ω) be a labeled poset. A (P, ω)-partition is a function σ : P → {0, 1, 2,... } such that
• σ(x) ≥ σ(y) if x ≤P y, • σ(x) > σ(y) if x ≤P y and ω(x) > ω(y). For a (P, ω)-partition σ, the size of σ is defined by X |σ| = σ(x). x P ∈ Definition 2.9. A linear extension of P is an arrangement (t1, t2, . . . , tn) of the elements in P such that if ti
Definition 2.10. The Jordan-H¨olderset L(P, ω) of P is the set of permutations of the form ω(t1)ω(t2) ··· ω(tn) for some linear extension (t1, t2, . . . , tn) of P . It is well known [22, Theorem 3.15.7] that P qmaj(π) X σ π (P,ω) (2) q| | = ∈L , (q; q) σ n where the sum is over all (P, ω)-partitions σ. We next define a polytope obtained from a poset in a natural way. For simplicity, we will use n the same letter xi for the elements xi of a poset P , the coordinates of R , and also the integration variables.
Definition 2.11. Let P be a poset on {x1, . . . , xn}. For an n-dimensional box
I = {(x1, . . . , xn): ai ≤ xi ≤ bi}, the truncated order polytope of P inside I is defined by
OI (P ) = {(x1, . . . , xn) ∈ I : xi ≤ xj if xi ≤P xj}. The order polytope of P is defined by
O(P ) = O[0,1]n (P ). An important special case of order polytopes is a simplex, which is an order polytope of a chain. Let us first define chains and anti-chains.
Definition 2.12. Let P be a poset. Two elements x and y are called comparable if x ≤P y or y ≤P x, and incomparable otherwise. A chain is a poset in which any two elements are comparable. An anti-chain is a poset in which any two distinct elements are incomparable.
Definition 2.13. For π = π1 ··· πn ∈ Sn, we denote by Pπ the chain on {x1, . . . , xn} with relations xπ1 < xπ2 < ··· < xπn . For real numbers a < b, we call n n O[a,b] (Pπ) = {(x1, . . . , xn) ∈ R : a ≤ xπ1 ≤ · · · ≤ xπn ≤ b} the truncated simplex.
Note that O[0,1]n (Pπ) = O(Pπ) is the standard simplex which corresponds to a permutation π and has volume 1/n!. We end this section with definitions for partitions and Schur functions.
Definition 2.14. A partition is a weakly decreasing sequence λ = (λ1, . . . , λn) of non-negative integers. Each λi is called a part of λ. The length `(λ) of λ is the number of nonzero parts. We identify a partition λ with its Young diagram
λ = {(i, j) : 1 ≤ i ≤ `(λ), 1 ≤ j ≤ λi}.
The transpose λ0 of a partition λ is defined by
λ0 = {(j, i) : 1 ≤ i ≤ `(λ), 1 ≤ j ≤ λi}.
m1 m2 If λ has mi parts equal to i for i ≥ 1, we also write λ as (1 , 2 ,... ). For two partitions λ = (λ1, . . . , λn) and µ = (µ1, . . . , µn) we define
λ + µ = (λ1 + µ1, λ2 + µ2, . . . , λn + µn). We also define δn = (n − 1, n − 2,..., 1, 0).
Definition 2.15. Let Parn denote the set of all partitions with length at most n. For a partition λ ∈ Parn, we denote n X b(λ) = (i − 1)λi, i=1 and qλ = (qλ1 , . . . , qλn ). 6 JANG SOO KIM AND DENNIS STANTON
Definition 2.16. For a partition λ = (λ1, . . . , λn), the alternant aλ(x1, . . . , xn) is defined by
λi+n i n aλ(x1, . . . , xn) = det(xj − )i,j=1.
Definition 2.17. For a partition λ = (λ1, . . . , λn), the Schur function sλ(x1, . . . , xn) is defined by
aλ+δn (x1, . . . , xn) sλ(x1, . . . , xn) = . aδn (x1, . . . , xn) Remark 2.18. Note that denominator of the Schur function is the Vandermonde determinant Y aδn (x1, . . . , xn) = (xi − xj) = ∆(x). 1 i Similarly the second integral on the left side is Z X∞ j j j L2 = (1 − q) bf(x, bq ) − xf(x, xq ) q dqx a x b ≤ ≤ j=0 X∞ = (1 − q)2 b2qi+jf(bqi, bqj) − b2q2i+jf(bqi, bqi+j) i,j=0 − abqi+jf(aqi, bqj) − a2q2i+jf(aqi, bqi+j) . So the difference is 2 X∞ 2 i+2j i+j j i+j i j 2i+j i i+j L1 − L2 = (1 − q) b q f(bq , bq ) − q f(bq , bq ) + q f(bq , bq ) i,j=0 − a2 qi+2jf(aqi+j, aqj) − qi+jf(aqi, aqj) + q2i+jf(aqi, aqi+j) . Let’s check when a term qs+tf(bqs, bqt) occurs in the first three terms, for non-negative integers s and t. The first term allows s ≥ t, the third term s ≤ t, while the second term is all s, t. So the remaining terms are Z b 2 X∞ 2 2s s s 2 2s s s (1 − q) b q f(bq , bq ) − a q f(aq , aq ) = (1 − q) xf(x, x)dqx. s=0 a Note that Proposition 3.3 implies that if f(x, y) vanishes on the boundary x = y, then we can exchange the order of integration Z Z f(x, y)dqxdqy = f(x, y)dqydqx. a x y b a x y b ≤ ≤ ≤ ≤ ≤ ≤ We show that this is also true for certain polytopes, see Corollary 3.5. Proposition 3.4 gives a general expansion to evaluate q-integrals over special polytopes. Proposition 3.4. Suppose that S is a set of pairs (i, j) with 1 ≤ i =6 j ≤ n. For each element (i, j) ∈ S, let ti,j be an integer. For fixed integers ri ≥ si ≥ 0, let ri si ti,j Q = {q ≤ xi ≤ q : 1 ≤ i ≤ n} ∩ {q xi ≤ xj :(i, j) ∈ S}. Then for π = π1 . . . πn ∈ Sn we have Z n X k1 kn k1+ +kn f(x1, . . . , xn)dqxπ1 ··· dqxπn = (1 − q) f(q , . . . , q )q ··· , Q k1,...,kn where the sum is over all integers k1, . . . , kn satisfying • si ≤ ki < ri for 1 ≤ i ≤ n, 1 1 • ti,j + ki ≥ kj if (i, j) ∈ S and π− (i) < π− (j), 1 1 • ti,j + ki > kj if (i, j) ∈ S and π− (i) > π− (j). Proof. Let D be the set of points in Rn satisfying the inequalities in Q. By definition, the left hand side is Z max(Dn) Z max(D1) (3) ··· f(x1, . . . , xn)dqxπ1 ··· dqxπn , min(Dn) min(D1) where Di = Di(xπi+1 , . . . , xπn ) = {yπi :(y1, . . . , yn) ∈ D, yπj = xπj for j > i}. Note that Di is the set of all real numbers yπi such that rπi sπi • q ≤ yπi ≤ q , tπi,πj • q yπi ≤ xπj for all j > i with (πi, πj) ∈ S, 8 JANG SOO KIM AND DENNIS STANTON tπj ,πi • q xπj ≤ yπi for all j > i with (πj, πi) ∈ S. Thus t sπi πi,πj max(Di) = min{q , q− xπj : j > i, (πi, πj) ∈ S}, t rπi πj ,πi min(Di) = max{q , q xπj : j > i, (πi, πj) ∈ S}. kπj ai bi In other words, if xπj = q for j = i + 1, . . . , n, then max(Di) = q and min(Di) = q , where ai = ai(kπi+1 , . . . , kπn ) = max({sπi } ∪ {kπj − tπi,πj : j > i, (πi, πj) ∈ S}), bi = bi(kπi+1 , . . . , kπn ) = min({rπi } ∪ {kπj + tπj ,πi : j > i, (πj, πi) ∈ S}). Thus, by Lemma 3.1, (3) is equal to b 1 b 1 n− 1− n X X k k k + +k (1 − q) ··· f(q 1 , . . . , q n )q 1 ··· n . kπn =an kπ1 =a1 Note that once kπi+1 , . . . , kπn are determined, we have ai ≤ kπi < bi if and only if • sπi ≤ kπi < rπi , • kπj − tπi,πj ≤ kπi for all j > i with (πi, πj) ∈ S, • kπi < kπj + tπj ,πi for all j > i with (πj, πi) ∈ S. One can easily check that the above sum is equivalent to the right hand side of the equation in this proposition. An immediate corollary of Proposition 3.4 is that if f(x1, . . . , xn) vanishes on the boundary ti,j q xi = xj for all (i, j) ∈ S, then we can change the order of integration. Corollary 3.5. We follow the same notation in Proposition 3.4. Suppose that the function f ti,j satisfies that f(x1, . . . , xn) = 0 if q xi = xj for any pair (i, j) ∈ S. Then for any permutations π, σ ∈ Sn, we have Z Z f(x1, . . . , xn)dqxπ1 ··· dqxπn = f(x1, . . . , xn)dqxσ1 ··· dqxσn . Q Q Corollary 3.6 gives a sufficient condition for changing the domain of a q-integral from one standard simplex to another. We will use this corollary later in this paper. Corollary 3.6. Suppose that f(x1, . . . , xn) is symmetric in x1, . . . , xn and f(x1, . . . , xn) = 0 if xi = xj for any i =6 j. Then for π ∈ Sn and 0 < q < 1, we have Z Z f(x1, . . . , xn)dqx1 ··· dqxn = f(x1, . . . , xn)dqx1 ··· dqxn. 0 x x 1 0 x x 1 ≤ π1 ≤···≤ πn ≤ ≤ 1≤···≤ n≤ 1 Proof. Let σ = π− . By renaming the variables xi 7→ xσi , the left hand side becomes Z f(xσ1 , . . . , xσn )dqxσ1 ··· dqxσn . 0 x x 1 ≤ 1≤···≤ n≤ Since f is symmetric, this is equal to Z f(x1, . . . , xn)dqxσ1 ··· dqxσn . 0 x x 1 ≤ 1≤···≤ n≤ We finish the proof by applying Corollary 3.5. Corollary 3.7 allows us to change the order of integration by modifying the polytope. ti,j Corollary 3.7. Let Q be a family of inequalities of the form q xi ≤ xj for 1 ≤ i =6 j ≤ n and ri si the inequalities q ≤ xi ≤ q for 1 ≤ i ≤ n. Then for π, σ ∈ Sn we have Z Z f(x1, . . . , xn)dqxπ1 ··· dqxπn = f(x1, . . . , xn)dqxσ1 ··· dqxσn , Q Q0 where Q0 is the family of inequalities obtained from Q as follows. ON q-INTEGRALS OVER ORDER POLYTOPES 9 ti,j 1 1 1 1 • If Q contains the inequality q xi ≤ xj such that π− (i) > π− (j) and σ− (i) < σ− (j), ti,j 1 then replace this inequality by q − xi ≤ xj. ti,j 1 1 1 1 • If Q contains the inequality q xi ≤ xj such that π− (i) < π− (j) and σ− (i) > σ− (j), ti,j +1 then replace this inequality by q xi ≤ xj. • The remaining inequalities of Q are unchanged. Proof. This is proved by expanding both sides using Proposition 3.4. 4. q-integrals over order polytopes In this section we consider q-integrals over order polytopes of posets. The main result (Theo- rem 4.4) is that the q-volume of an order polytope of a poset may be written, up to a factor, as the maj-generating function of the linear extensions of that poset. Thus these q-integrals may be evaluated using permutation enumeration. Recall the truncated order polytope OI (P ) in Definition 2.11. We first use Proposition 3.4 to evaluate an arbitrary integral over OI (P ) as a sum over restricted (P, ω)-partitions σ. Theorem 4.1. Let P be a poset on {x1, . . . , xn} and ωn : P → [n] be the labeling of P given by ωn(xi) = i for 1 ≤ i ≤ n. For integers r1, r2, . . . , rn, s1, s2, . . . , sn with ri ≥ si ≥ 0, let ri si I = {(x1, . . . , xn): q ≤ xi ≤ q }. Then Z n X σ(x1) σ(xn) σ f(x1, . . . , xn)dqx1 ··· dqxn = (1 − q) f(q , . . . , q )q| |, (P ) OI σ where the sum is over all (P, ωn)-partitions σ satisfying si ≤ σ(xi) < ri for 1 ≤ i ≤ n. Proof. This is obtained immediately from Proposition 3.4 by taking π = 12 ··· n, S = {(i, j): xi ≤P xj}, ti,j = 0. Note that in Theorem 4.1 the labeling of the poset is closely related to the order of integration. The next corollary expresses a q-integral over a truncated order polytope as a sum of q-integrals of truncated simplices. Corollary 4.2. Let P be a poset on {x1, . . . , xn} and ωn : P → [n] the labeling of P given by ωn(xi) = i for 1 ≤ i ≤ n. For integers r1, r2, . . . , rn, s1, s2, . . . , sn with ri ≥ si ≥ 0, let ri si I = {(x1, . . . , xn): q ≤ xi ≤ q }. Then Z X Z f(x1, . . . , xn)dqx1 ··· dqxn = f(x1, . . . , xn)dqx1 ··· dqxn, I (P ) π (P,ω ) I (Pπ ) O ∈L n O Pπ is the chain xπ1 ≤ · · · ≤ xπn . Proof. This can be proved by the standard arguments in the (P, ω)-partition theory, see [22, Lemma 3.15.3]. We shall need the following lemma later. Lemma 4.3. Let f(x1, . . . , xn) be a function such that f(x1, . . . , xn) = 0 if xi = xj for any i =6 j. Then Z X µ+δ µ+δ 1 q| n|f(q n ) = f(x , . . . , x )d x ··· d x . (1 − q)n 1 n q 1 q n µ Par 0 x1 xn 1 ∈ n ≤ ≤···≤ ≤ Proof. By Theorem 4.1 and the assumption on the function f(x1, . . . , xn), the right side is equal to X i i i + +i X i i i + +i f(q 1 , . . . , q n )q 1 ··· n = f(q 1 , . . . , q n )q 1 ··· n . i i i 0 i >i > >i 0 1≥ 2≥···≥ n≥ 1 2 ··· n≥ This is equal to the left side. 10 JANG SOO KIM AND DENNIS STANTON n By taking I = [0, 1] and f(x1, . . . , xn) = 1 in Theorem 4.1, we obtain that the q-volume of the order polytope O(P ) is the generating function for (P, ω)-partitions. The equivalence of the two equations in Theorem 4.4 follows from the well known fact (2) on P -partition theory. Theorem 4.4. [q-volume of order polytope] Let P be a poset on {x1, . . . , xn} with labeling ωn given by ωn(xi) = i. Then Z n X σ Vq(O(P )) = dqx1 ··· dqxn = (1 − q) q| |, (P ) O σ where the sum is over all (P, ωn)-partitions σ. Equivalently, Z 1 X maj(π) Vq(O(P )) = dqx1 ··· dqxn = q . [n]q! (P ) π (P,ω ) O ∈L n As a special case of Theorem 4.4, let us consider the anti-chain P on {1, 2, . . . , n}. Then O(P ) is the n-dimensional unit cube whose q-volume is 1 and L(P, ω) = Sn. If we apply Theorem 4.4 to P , we obtain 1 X 1 = qmaj(π). [n]q! π S ∈ n This is a well known result for the maj-generating function for permutations, see [22]. Let’s consider another special case of order polytopes, which are truncated simplices. The following lemma will be used to evaluate the q-volume of a truncated simplex. Lemma 4.5. Let π ∈ Sn and r > s ≥ 0. Then r s des(π) X i + +i sn+maj(π) (q − − ; q)n q 1 ··· n = q . (q; q)n r>i1 in s i >i ≥···≥if j Des(≥ π) j j+1 ∈ Proof. Let A be the set of partitions λ = (λ1, . . . , λn) such that r > λ1 ≥ · · · ≥ λn ≥ s and λj > λj+1 if j ∈ Des(π). Observe that if λ ∈ A, then considering the Young diagram of the s r 1 transpose λ0 of λ we have (n ) ⊆ λ0 ⊆ (n − ) and j ∈ λ0 for j ∈ Des(π). Thus the left hand side is equal to r s des(π) X λ sn maj(π) r − s − 1 − des(π) + n sn maj(π) (q − − ; q)n q| | = q q = q q , n (q; q)n λ A q ∈ which finishes the proof. We now have a formula for the q-volume of a truncated simplex. This will be used later to evaluate the q-beta integral and give a combinatorial interpretation for the q-Selberg integral. Corollary 4.6. [q-volume of a truncated simplex] For π ∈ Sn and real numbers a < b, the q-volume of the truncated simplex O[a,b]n (Pπ) is Z n maj(π) b q des(π) Vq(O[a,b]n (Pπ)) = dqx1 ··· dqxn = (aq− /b; q)n. a x x b [n]q! ≤ π1 ≤···≤ πn ≤ Proof. Since both sides are polynomials in a and b, it is sufficient to show the following for integers r > s ≥ 0: Z sn+maj(π) q r s des(π) dqx1 ··· dqxn = (q − − ; q)n. qr x x qs [n]q! ≤ π1 ≤···≤ πn ≤ This follows from Theorem 4.1 and Lemma 4.5. By considering the q-volume of the box [a, 1]n, we obtain an identity for a generating function for permutations with maj and des statistics. Corollary 4.7. For a non-negative integer n, we have X maj(π) des(π) n q (aq− ; q)n = (1 − a) [n]q!. π S ∈ n ON q-INTEGRALS OVER ORDER POLYTOPES 11 n Proof. Let I = [a, 1] and P the anti-chain on {x1, . . . , xn}. Then OI (P ) = I and Vq(OI (P )) = (1 − a)n. On the other hand, by Corollary 4.2 and Corollary 4.6, we have maj(π) n X X q des(π) (1 − a) = Vq(OI (P )) = Vq(O[a,1]n (Pπ)) = (aq− ; q)n. [n]q! π S π S ∈ n ∈ n By multiplying both sides by [n]q!, we have the stated result. Theorem 4.8 is another generating function for permutations with maj and des statistics due to MacMahon. This result is often called Carlitz’s formula [6], see [10, p. 6]. Theorem 4.8. We have X des(π) maj(π) X n i t q = (t; q)n+1 [i + 1]q t . π S i 0 ∈ n ≥ des(π) We show that Corollary 4.6 and Theorem 4.8 are equivalent. By expanding (aq− ; q)n using the q-binomial theorem and comparing the coefficients of ak in both sides, one can restate Corollary 4.6 as follows: for 0 ≤ k ≤ n, X k des(π) maj(π) (k) n (4) (q− ) q = q− 2 (q; q)k(q; q)n k. k − π S ∈ n n i+1 n n On the other hand, by expanding the numerator of [i + 1]q = (1 − q ) /(1 − q) , using the binomial theorem and the geometric series, one can check that Theorem 4.8 can be restated as n X des(π) maj(π) 1 X n i i+1 (5) t q = (−q) (t; q)i(q t; q)n i. (1 − q)n i − π S i=0 ∈ n k If we substitute t = q− in (5) for 0 ≤ k ≤ n then we obtain (4). In order to obtain (5) from (4) k one can argue as follows. By (4), we know that (5) is true when t = q− for 0 ≤ k ≤ n. Since both sides of (5) are polynomials in t of degree at most n and the equation has n + 1 different solutions, both sides are the same as polynomials in t. When a = 0 and b = 1 in Corollary 4.6 we obtain the following corollary. It explicitly demon- strates the failure of Fubini’s theorem by finding the differences in the q-volumes of the standard n! simplices which lie inside an n-dimensional cube. Corollary 4.9. [q-volume of a simplex] For π ∈ Sn, the q-volume of the simplex O(Pπ) is Z qmaj(π) Vq(O(Pπ)) = dqx1 ··· dqxn = . 0 x x 1 [n]q! ≤ π1 ≤···≤ πn ≤ 5. Operations on posets Suppose that a poset P is modified to obtain another poset P 0. Is Vq(O(P 0)) related to the q-integral defining Vq(O(P ))? In this section we answer this question for the following types of modifications: (1) attaching a chain below an element (Lemma 5.1), (2) attaching a chain above an element (Lemma 5.2), (3) attaching a chain between two elements (Lemma 5.3), (4) inserting a chain which interlaces another chain (Lemmas 5.5 and 5.6). First we insert a chain below a fixed element. See the left figure in Figure 1. Lemma 5.1 (Scaredy Cat Lemma). Let P be a poset on {x1, . . . , xn} and t ∈ [n]. Let Q be the poset on {x1, . . . , xn, y1, . . . , ym} with relations xi ≤Q xj if and only if xi ≤P xj, and y1 ≤Q · · · ≤Q ym ≤Q xt. In other words, Q is obtained from P by attaching a chain y1 ≤ · · · ≤ ym below xt. Then we have Z Z m xt f(x1, . . . , xn)dqx1 ··· dqxn = [m]q! f(x1, . . . , xn)dqy1 ··· dqymdqx1 ··· dqxn. (P ) (Q) O O 12 JANG SOO KIM AND DENNIS STANTON ym xt yρm . . . xt y2 ... ym y .. y1 ρ2 . xs y P x ρ1 y2 s P y1 P Figure 1. Illustrations of the scaredy cat lemma (left), the happy cat lemma (middle), and the attaching chain lemma (right). Proof. By Corollary 4.6, we have Z m xt dqy1 ··· dqym = . 0 y y x [m]q! ≤ 1≤···≤ m≤ t Thus, the left hand side of the equation can be written as Z Z [m]q! dqy1 ··· dqym f(x1, . . . , xn)dqx1 ··· dqxn. (P ) 0 y y x O ≤ 1≤···≤ m≤ t This is equal to the right hand side of the equation. Next we insert a chain above a fixed element. See the middle figure in Figure 1. Lemma 5.2 (Happy Cat Lemma). Let P be a poset on {x1, . . . , xn} and s ∈ [n]. Let Q be the poset on {x1, . . . , xn, y1, . . . , ym} with relations xi ≤Q xj if and only if xi ≤P xj, and ≤Q xs ≤ y1 ≤Q · · · ≤Q ym. In other words, Q is obtained from P by attaching a chain y1 ≤ · · · ≤ ym above xs. Then we have Z Z (qxs; q)mf(x1, . . . , xn)dqx1 ··· dqxn = [m]q! f(x1, . . . , xn)dqx1 ··· dqxndqy1 ··· dqym. (P ) (Q) O O Proof. By Corollary 4.6, we have Z (qxs; q)m dqy1 ··· dqym = . qx y y 1 [m]q! s≤ 1≤···≤ m≤ Thus, the left hand side of the equation can be written as Z Z [m]q! dqy1 ··· dqym f(x1, . . . , xn)dqx1 ··· dqxn (P ) qxs y1 ym 1 ZO ≤ ≤···≤ ≤ =[m]q! f(x1, . . . , xn)dqy1 ··· dqymdqx1 ··· dqxn, D where D is the set of inequalities given by D = {0 ≤ xi ≤ xj ≤ 1 : xi ≤P xj} ∪ {0 ≤ y1 ≤ · · · ≤ ym ≤ 1} ∪ {qxs ≤ yi : i ∈ [m]}. By Corollary 3.7, we have Z Z f(x1, . . . , xn)dqy1 ··· dqymdqx1 ··· dqxn = f(x1, . . . , xn)dqx1 ··· dqxndqy1 ··· dqym, D D0 where D0 = {0 ≤ xi ≤ xj ≤ 1 : xi ≤P xj} ∪ {0 ≤ y1 ≤ · · · ≤ ym ≤ 1} ∪ {xs ≤ yi : i ∈ [m]}. Since D0 and O(Q) represent the same domain, we get the lemma. Now we state but do not prove the attaching chain lemma, since the proof is similar to the previous proofs. See the right figure in Figure 1. ON q-INTEGRALS OVER ORDER POLYTOPES 13 yn yn xn xn 1 − ...... x2 y2 y2 x2 x1 y1 y1 x1 Figure 2. Illustrations of x ≺ y for x = (x1, . . . , xn 1), y = (y1, . . . , yn) on the − left and x ≺ y for x = (x1, . . . , xn), y = (y1, . . . , yn) on the right. Lemma 5.3 (Attaching chain lemma). Let ρ ∈ Sm. Let P be a poset on {x1, . . . , xn} with xs ≤P xt. Define Q to be the poset on {x1, . . . , xn, y1, . . . , ym} with relations xi ≤Q xj if and only if xi ≤P xj, and xs ≤Q yρ1 ≤Q · · · ≤Q yρm ≤Q xt. Then, we have Z maj(ρ) m des(ρ) (6) q xt (q− xs/xt; q)mf(x1, . . . , xn)dqx1 ··· dqxn (P ) O Z = [m]q! f(x1, . . . , xn)dqy1 ··· dqymdqx1 ··· dqxn, (Q) O Z maj(ρ) m 1 des(ρ) (7) q xt (q − xs/xt; q)mf(x1, . . . , xn)dqx1 ··· dqxn (P ) O Z = [m]q! f(x1, . . . , xn)dqx1 ··· dqxsdqy1 ··· dqymdqxs+1 ··· dqxn. (Q) O We note that (7) holds if the order of integration on the right hand side is obtained from dqx1 ··· dqxn by inserting dqy1 ··· dqym anywhere between xs and xt. There is another way of attaching a chain. We need some definitions. Definition 5.4. For two sets of variables x = (x1, x2, . . . , xn 1) and y = (y1, y2, . . . , yn) with y having one more variable than x, we denote by x ≺ y the set of− inequalities {y1 ≤ x1 ≤ y2 ≤ x2 ≤ · · · ≤ yn 1 ≤ xn 1 ≤ yn}. − − Moreover, if x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn) have the same number of variables, we also denote x ≺ y the set of inequalities {x1 ≤ y1 ≤ x2 ≤ y2 ≤ · · · ≤ xn ≤ yn}. See Figure 2. Lemma 5.5 (Interlacing chain lemma 1). Let P be a poset on {x1, . . . , xN }. Suppose that y = (y1 y1 Lemma 5.6 (Interlacing chain lemma 2). Let P be a poset on {x1, . . . , xN }. Suppose that y = (y1 z1 Lemma 5.7. Let z = (z1, z2, . . . , zn 1) and y = (y1, y2, . . . , yn). If `(λ) < n, we have − n 1 Y− Z sλ(y)∆(y) = [λi + n − i]q sλ(z)∆(z)dqz1 ··· dqzn 1. z y − i=1 ≺ Proof. We note that the integrand on the right side is a determinant. Applying the q-integrals gives another determinant, which is the left side. Specifically, n−1 λ +n 1 j ( 2 ) j sλ(z1, . . . , zn 1)∆(z) = (−1) det zi − − . − 1 i,j n 1 ≤ ≤ − Evaluating the q-integral on zi from yi to yi+1 evaluates the right side of Lemma 5.7 as n−1 λ +n j λ +n j ( 2 ) j j (−1) det yi+1 − − yi − . 1 i,j n 1 ≤ ≤ − For the left side, n λ +n j (2) j sλ(y1, . . . , yn)∆(y) = (−1) det yi − 1 i,j n ≤ ≤ th th has an n column of all 1’s because λn = 0. By successively subtracting the i row from the i + 1th row for i = n − 1 to i = 1 we obtain a final column of (1, 0,..., 0)T . Expanding the determinant along this column gives n λ +n j λ +n j (2) n 1 j j sλ(y1, . . . , yn)∆(y) =(−1) (−1) − det yi+1 − − yi − 1 i,j n 1 ≤ ≤ − n−1 λ +n j λ +n j ( 2 ) j j =(−1) det yi+1 − − yi − 1 i,j n 1 ≤ ≤ − which is the right side. The version of Lemma 5.7 when λn > 0 is next. Lemma 5.8. Let z = (z1, z2, . . . , zn) and y = (y1, y2, . . . , yn). If `(λ) = n, we have n Y Z sλ(y)∆(y) = [λi + n − i]q sλ (1n)(z)∆(z)dqz1 ··· dqzn. z y − i=1 ≺ Proof. Let u = (u1, u2, . . . , un+1) = (0, y1, y2, . . . , yn). By Lemma 5.7, we have n Y Z sλ (1n)(u1, . . . , un+1)∆(u) = [(λi − 1) + (n + 1) − i]q sλ (1n)(z1, . . . , zn)∆(z)dqz1 ··· dqzn. − z u − i=1 ≺ Since sλ (1n)(u1, . . . , un+1) = sλ (1n)(y1, . . . , yn) = sλ(y1, . . . , yn)/y1 ··· yn − − and ∆(u) = y1 ··· yn∆(y) we are done. ON q-INTEGRALS OVER ORDER POLYTOPES 15 6. Examples of q-integrals In this section we use the constructions in Section 4 to evaluate the q-integrals. This includes the q-beta integral, a q-analogue of Dirichlet’s integral, and a general q-beta integral due to Andrews and Askey [1]. We will then find a connection with linear extensions of forest posets. 6.1. The q-beta integral. The following is the well known integral called the q-beta integral. We now prove this using our methods. The idea is to add two chains to a point: one chain below it and the other chain above it, naturally labeled. The resulting poset is again a chain whose q-volume is easily computed. Corollary 6.1. We have Z 1 n [n]q![m]q! x (xq; q)mdqx = . 0 [n + m + 1]q! Proof. By (5.2) and (5.1) we have Z 1 Z n x (xq; q)mdqx = [n]q![m]q! dqy1 ··· dqyndqxdqz1 ··· dqzm. 0 0 y y x z z 1 ≤ 1≤···≤ n≤ ≤ 1≤···≤ m≤ By Corollary 4.9, we get the q-beta integral formula. 6.2. A q-analog of Dirichlet integral. We now consider the simplex n Ωn = {(x1, . . . , xn) ∈ [0, 1] : x1 + ··· + xn ≤ 1}. Dirichlet’s integral is the following, see [2, Theorem 1.8.6]: Z α 1 α 1 a 1 Γ(α1) ··· Γ(αn+1) (8) x 1− ··· x n (1 − x − · · · − x ) n+1 dx ··· dx = . 1 n − 1 n − 1 n ··· Ωn Γ(α1 + + αn+1) By introducing new variables yi = x1 + ··· + xi and integers ki = αi − 1, we get an equivalent version of (8) Z k1 k2 kn kn+1 (9) y1 (y2 − y1) ··· (yn − yn 1) (1 − yn) dy1 ··· dyn 0 y y y 1 − ≤ 1≤ 2≤···≤ n≤ k ! ··· k ! = 1 n+1 . (n + k1 + ··· + kn+1)! A q-analogue of (9) is given by the next corollary. It generalizes Corollary 6.1 which is the case n = 1. Corollary 6.2. For nonnegative integers k1, . . . , kn+1, we have Z n ··· k1 Y ki [k1]q! [kn+1]q! y1 (qyn; q)kn+1 yi (qyi 1/yi; q)ki dqy1 ··· dqyn = . 0 y y y 1 − [n + k1 + ··· + kn+1]q! ≤ 1≤ 2≤···≤ n≤ i=2 Proof. We generalize the proof of Corollary 6.1. We start with a chain y1 ≤ y2 ≤ · · · ≤ yn. Attach a chain with ki elements between yi 1 and yi, for 2 ≤ i ≤ n, and also a chain with k1 elements − below y1, and a chain with kn+1 elements above yn. We obtain a chain with n + k1 + ··· + kn+1 elements which is naturally labeled. By applying the scaredy cat lemma once, the second part (7) in the attaching chain lemma with ρ the identity permutation n − 1 times, and the happy cat lemma once, we obtain the left side multiplied by some q-factorials. The right side results from Corollary 4.9 with π being the identity. 16 JANG SOO KIM AND DENNIS STANTON 6.3. The general q-beta integral of Andrews and Askey. Andrews and Askey [1] generalized the q-beta integral as follows: for |q| < 1, Z b (qx/a; q) (qx/b; q) (1 − q)(q; q) (AB; q) ab(a/b; q) (b/a; q) (10) ∞ ∞ dqx = ∞ ∞ ∞ ∞ . a (Ax/a; q) (Bx/b; q) (A; q) (B; q) (a − b)(Ba/b; q) (Ab/a; q) ∞ ∞ ∞ ∞ ∞ ∞ In this subsection, by computing the q-volume of a truncated simplex in two different ways, we will show the following proposition which is equivalent to the special case of (10) with substitution r k k +1 r+1 s+1 (a, b, A, B) 7→ (aq − 1 , bq 2 , q , q ). Proposition 6.3. Let n, r, s, k1, k2 be nonnegative integers such that n = r+s+1, k1 ≤ r, k2 ≤ s, and k = k1 + k2 + 1 if s ≥ 1 and k = k1 if s = 0. Then Z b r k1 k2 [r]q![s]q! r+1 (k k1)(r+1) k x (aq− /x; q)r(xq− /b; q)sdqx = b q − (aq− /b; q)n. a [n]q! Proof. Let π ∈ Sn. By Corollary 4.6 the q-volume of the truncated simplex O[a,b]n (Pπ) is Z n maj(π) b q des(π) Vq(O[a,b]n (Pπ)) = dqx1 ··· dqxn = (aq− /b; q)n. a x x b [n]q! ≤ π1 ≤···≤ πn ≤ Now we compute this q-volume in a different way by decomposing the chain xπ1 ≤ · · · ≤ xπn into two chains. First we decompose π into π = σnτ using the largest integer n. Suppose that σ and τ have r and s letters respectively and des(σ) = k1, des(τ) = k2. Then n = r + s + 1 and k = k1 + k2 + 1 if k2 ≥ 1 and k = k1 if k2 = 0. The q-volume of O[a,b]n (Pπ) can be written as Z b Z Z ! dqy1 ··· dqyr dqz1 ··· dqzs dqxn, a a x x x x x x b ≤ σ1 ≤···≤ σr ≤ n n≤ τ1 ≤···≤ τs ≤ where y1, . . . , yr and z1, . . . , zs are obtained by rearranging xσ1 , . . . , xσr and xτ1 , . . . , xτs respec- tively so that subscripts are increasing. By applying Corollary 4.6 to the two inside integrals, the above is equal to Z b r maj(σ) s maj(τ) x q k1 b q k2 (11) (aq− /x; q)r (xq− /b; q)sdqx. a [r]q! [s]q! Note that maj(π) = maj(σ) + maj(τ) + (r + 1)(k2 + 1) if s ≥ 1 and maj(π) = maj(σ) if s = 0. In either case we can write maj(π) = maj(σ)+maj(τ)+(r+1)(k −k1). This completes the proof. We now consider the case s ≥ 1 in Proposition 6.3 so that k = k1 + k2 + 1. One can rewrite the integral in Proposition 6.3 as b 1 r+k k r Z (xq 1 /a; q) (xq 2 /b; q) r r ( ) k1r − − (−1) a q 2 − ∞ ∞ dqx k1+1 s k2 a (xq /a; q) (xq − /b; q) ∞ ∞ bqk2+1 1 r+k k r Z (xq 1 /a; q) (xq 2 /b; q) r r ( ) k1r − − = (−1) a q 2 − ∞ ∞ dqx, k1+1 s k2 aqr−k1 (xq /a; q) (xq − /b; q) ∞ ∞ j where the equality follows from the fact that the integrand is 0 if x = bq for 0 ≤ j ≤ k2 and j x = aq for 0 ≤ j ≤ r − k1 − 1. Thus Proposition 6.3 is equivalent to k2+1 r Z bq 1 r+k1 k2 r r+1 kr+k k1 ( ) k (xq − /a; q) (xq− /b; q) (−1) b [r]q![s]q!q − − 2 (aq− /b; q)n ∞ ∞ dqx = . k1+1 s k2 r aqr−k1 (xq /a; q) (xq − /b; q) a [n]q! ∞ ∞ r k k +1 r+1 s+1 This is the (a, b, A, B) 7→ (aq − 1 , bq 2 , q , q ) case of (10). ON q-INTEGRALS OVER ORDER POLYTOPES 17 x5 x9 x6 x3 x4 x7 x8 x1 x2 Figure 3. A forest poset. x5 x6 x9 x3 x4 x8 x7 x2 x1 Figure 4. The forest poset Fa, where F is the forest in Figure 3 and a = (a1, . . . , a9). 6.4. q-integrals of monomials over the order polytope of a forest poset. In this subsection we consider special posets called forests and evaluate the q-integral of a monomial over the order polytope coming from these posets. These forests have an order polytope whose q-volume has a hook formula, see Corollary 6.9. Definition 6.4. A poset is called a forest if every element is covered by at most one element. For a forest poset F and its element x, the hook length hF (x) of x is defined to be the number of elements y ∈ F such that y ≤F x. An element of a forest is called isolated if it has no relation with other elements. A leaf is a non-isolated element whose hook length is 1. Example 6.5. Let F be the forest in Figure 3. Then x6 is an isolated element and x1, x2, x4, x7, x8 are leaves. These elements have hook length 1. The hook lengths of other elements are hF (x3) = 3, hF (x5) = 5, hF (x9) = 3. Definition 6.6. For a forest F and a sequence a = (a1, . . . , an) of nonnegative integers, let Fa be the poset obtained from F by attaching ai leaves to xi for each i ∈ [n], see Figure 4. Example 6.7. Let F be the poset in Figure 3 and a = (a1, . . . , a9) = (0, 3, 2, 1, 2, 3, 0, 2, 1). The forest Fa is shown in Figure 4, where the short edges are newly added from F in Figure 3. We have hFa (x1) = 1, hFa (x2) = 4, hFa (x3) = 8, hFa (x4) = 2, hFa (x5) = 13, hFa (x6) = 4, hFa (x7) = 1, hFa (x8) = 3, hFa (x9) = 6. The next corollary allows us to evaluate any multiple q-integral of a monomial over the order polytope of a naturally labeled forest. Theorem 6.8. Let F be a forest on {x1, . . . , xn} with labeling ωn given by ωn(xi) = i. Suppose that ωn is a natural labeling. Let a = (a1, . . . , an) be a sequence of nonnegative integers. Then, we have Z a1 an Y 1 x1 ··· xn dqx1 ··· dqxn = . [hF (v)]q (F ) v F a O ∈ a Proof. We prove this using induction on n. If n = 1, then both sides are equal to 1/[a + 1]q. Suppose that n > 1 and the theorem is true for n − 1. Since ω is a natural labeling, x1 is an isolated element or a leaf in F . 18 JANG SOO KIM AND DENNIS STANTON Case 1: x1 is an isolated element in F . Then the range for x1 in the q-integral is 0 ≤ x1 ≤ 1. Let F 0 = F − {x1} and a0 = (a2, . . . , an). By the induction hypothesis, we have Z Z 1 Z a1 an a1 a2 an x1 ··· xn dqx1 ··· dqxn = x1 dqx1 x2 ··· xn dqx2 ··· dqxn (F ) 0 (F 0) O O 1 Y 1 Y 1 = = . 0 [a1 + 1]q 0 [hF 0 (v)]q [hFa (v)]q v F a v Fa ∈ a0 ∈ Case 2: x1 is a leaf in F . Let xk be the unique element covering x1. Then the range for x1 in the q-integral is 0 ≤ x1 ≤ xk. Let F 0 = F − {x1} and a0 = (a20 , . . . , an0 ), where ai0 = ai if i =6 k and ak0 = ak + a1 + 1. By the induction hypothesis, we have Z Z Z xk a1 an a1 a2 an x1 ··· xn dqx1 ··· dqxn = x1 dqx1 x2 ··· xn dqx2 ··· dqxn (F ) (F 0) 0 O O ! Z xa1+1 k a2 an = x2 ··· xn dqx2 ··· dqxn (F 0) [a1 + 1]q O Z 0 0 1 a2 an 1 Y 1 = x2 ··· xn dqx2 ··· dqxn = . [a1 + 1]q (F 0) [a1 + 1]q [hF 0 (v)]q v F 0 a0 O ∈ a0 It is easy to check that 1 Y 1 Y 1 = , 0 [a1 + 1]q 0 [hF 0 (v)]q [hFa (v)]q v F a v Fa ∈ a0 ∈ which completes the proof. If a = (0,..., 0) in Theorem 6.8 we obtain the q-volume of the order polytope of a forest. Using Theorem 4.4 we also obtain the maj-generating function for the linear extensions of a forest, which was first proved by Bj¨ornerand Wach [5]. Corollary 6.9. Let F be a forest on {x1, . . . , xn} with labeling ωn given by ωn(xi) = i. Suppose that ωn is naturally labeled. Then, we have 1 Vq(O(F )) = Q , v F [hF (v)]q ∈ X maj(π) [n]q! q = Q . [hF (v)]q π (F,ω ) v F ∈L n ∈ 7. q-Selberg integrals In this section we will find a combinatorial interpretation for a q-Selberg integral. We will use the first three lemmas of Section 4 to “insert” linear factors in q-integrals by building a “Selberg poset” whose q-volume is the q-Selberg integral, up to a constant. As a corollary of the q-Selberg integral evaluation, we obtain the factorization of the maj-generating function for the linear extensions, see Corollary 7.7. There are many generalizations of the Selberg integral, see [8]. We consider three q-Selberg integrals which are closely related. Definition 7.1. Let α β be complex numbers with Re(α) > 0, Re(β) > 0, and m a non-negative integer. The Askey-Selberg integral ASn(α, β, m) is defined by Z 1 Z 1 n Y α 1 Y 2m 1 m ··· xi − (qxi; q)β 1 xj q − xi/xj; q dqx1 ··· dqxn. − 2m 0 0 i=1 1 i The Kadell-Selberg integral KSn(α, β, m) is defined by Z 1 Z 1 n Y α 1 Y 2m 1 1 m ··· xi − (qxi; q)β 1 xj − q − xi/xj; q ∆(x)dqx1 ··· dqxn. − 2m 1 0 0 i=1 1 i The Askey-Kadell-Selberg integral AKSn(α, β, m) is defined by Z n Y α 1 Y 2m 1 1 m xi − (qxi; q)β 1 xj − q − xi/xj; q ∆(x)dqx1 ··· dqxn. − 2m 1 0 x1 xn 1 i=1 1 i It is easy to check that the integrand in KSn(α, β, m) is symmetric in the variables x1, x2, . . . , xn. Recall [2, p. 493] that the q-gamma function is defined by (q; q)x 1 − Γq(x) = x 1 . (1 − q) − Askey [3] conjectured that n αm n +2m2 n Y Γq(α + (j − 1)m)Γq(β + (j − 1)m)Γq(1 + jm) (12) AS (α, β, m) = q (2) (3) , n Γ (α + β + (n + j − 2)m)Γ (1 + m) j=1 q q which has been proved independently by Habsieger [11] and Kadell [13]. Kadell [14, Eq. (4.11)] showed that [n] m ! (13) AS (α, β, m) = q KS (α, β, m). n n! n More generally Kadell showed Lemma 7.2 which implies (13). Lemma 7.2. For an anti-symmetric function f(x), we have Z Z Y [n]p! f(x) (xi − pxj)dqx = f(x)∆(x)dqx. n n! n [a,b] 1 i Since the integrand of KSn(α, β, m) is symmetric under any permutation of x1, . . . , xn and zero whenever xi = xj, by Corollary 3.6 we have (14) KSn(α, β, m) = n! AKSn(α, β, m). By (12), (13) and (14), we have n αm n +2m2 n Y Γq(α + (j − 1)m)Γq(β + (j − 1)m)Γq(jm) (15) AKS (α, β, m) = q (2) (3) . n Γ (α + β + (n + j − 2)m)Γ (m) j=1 q q We will give a combinatorial interpretation of AKSn(α, β, m) when α − 1 = r, β − 1 = s and m are non-negative integers. This requires defining a poset, the Selberg poset, whose order polytope has q-volume given by the q-Selberg integral. Basically we start with a chain C of n elements. Insert two independent chains with m elements between any two elements of C. For each element of c ∈ C, insert a chain with s elements above c and a chain with r elements below c. Definition 7.3. We define the Selberg poset P (n, r, s, m) to be the poset in which the elements (a) (b) (k) are xi, yi , zi , wi,j for i, j ∈ [n], a ∈ [r], b ∈ [s], k ∈ [m] with i =6 j, and the covering relations are as follows: (1) (m) • xi < wi,j < ··· < wi,j < xj for 1 ≤ i < j ≤ n, (m) (1) • xi < wj,i < ··· < wj,i < xj for 1 ≤ i < j ≤ n, (1) (r) (1) (s) • yi < ··· < yi < xi < zi < ··· < zi for 1 ≤ i ≤ n. 20 JANG SOO KIM AND DENNIS STANTON We define W to be the following permutation of the elements of P (n, r, s, m): Y (1) (m) Y (1) (m) W = wi,j ··· wi,j wj,i ··· wj,i 1 i (1) (2) (3) (1) (2) (3) (1) (2) (3) (1) (2) (3) (1) (2) (3) (1) (2) (3) W = w1,2w1,2w1,2w1,3w1,3w1,3w1,4w1,4w1,4w2,3w2,3w2,3w2,4w2,4w2,4w3,4w3,4w3,4 (1) (2) (3) (1) (2) (3) (1) (2) (3) (1) (2) (3) (1) (2) (3) (1) (2) (3) w2,1w2,1w2,1w3,1w3,1w3,1w4,1w4,1w4,1w3,2w3,2w3,2w4,2w4,2w4,2w4,3w4,3w4,3 (1) (2) (1) (2) (1) (2) (1) (2) (1) (1) (1) (1) y1 y1 y2 y2 y3 y3 y4 y4 x1x2x3x4z1 z2 z3 z4 . The labeling ω of P (n, r, s, m) is shown in Figure 5. The following theorem implies that the Askey-Kadell-Selberg integral is a q-volume of the order polytope O(P (n, r, s, m)) up to a certain factor. Theorem 7.5. We have n Z Y Y r 2m 1 1 m ··· xi (qxi; q)s xj − q − xi/xj; q 2m 1 ∆(x)dqx1 dqxn 0 x1 xn 1 i=1 1 i n Z Y Y r 2m 1 1 m ··· xi (qxi; q)s xj − q − xi/xj; q 2m 1 ∆(x)dqx1 dqxn 0 x1 xn 1 i=1 1 i Proof. This is an immediate consequence of Theorem 7.5 and Theorem 4.4. ON q-INTEGRALS OVER ORDER POLYTOPES 21 (1) z4 52 x4 48 (1) (2) (3) 34 w4,3 y4 44 w3,4 18 (2) (1) (2) 35 w4,3 y4 43 w3,4 17 (1) (3) (1) (1) (3) 31 w4,2 36 w4,3 z3 51 w3,4 16 w2,4 15 (2) (2) 32 w4,2 x3 w2,4 14 47 (1) (3) (1) (2) (3) (1) (3) 25 w4,1 33 w4,2 28 w3,2 y3 42 w2,3 12 w2,4 13 w1,4 9 (2) (2) (1) (2) (2) 26 w4,1 29 w3,2 y3 41 w2,3 11 w1,4 8 (3) (1) (3) (1) (1) (3) (1) 27 w4,1 22 w3,1 30 w3,2 z2 50 w2,3 10 w1,3 6 w1,4 7 (2) (2) 23 w3,1 x2 w1,3 5 46 (3) (1) (2) (3) (1) 24 w3,1 19 w2,1 y2 40 w1,2 3 w1,3 4 (2) (1) (2) 20 w2,1 y2 39 w1,2 2 (3) (1) (1) 21 w2,1 z1 49 w1,2 1 x1 45 (2) y1 38 (1) y1 37 Figure 5. The Selberg poset P (n, r, s, m) for n = 4, r = 2, s = 1, m = 3 with labeling. Using the q-Selberg integral formula, we obtain an explicit product formula for the maj- generating function for linear extensions of the Selberg poset. Corollary 7.7. We have X qmaj(π) 1 π (P (n,r,s,m),ω) ∈L (m)(n)+(r+1)m(n)+2m2(n) n q 2 2 2 3 [N]q! Y [r + (j − 1)m]q![s + (j − 1)m]q![jm − 1]q! = n , n n 2(2) [r + s + 1 + (n + j − 2)m] ![m − 1] ! ([r]q!) ([s]q!) ([m]q!) j=1 q q where N and ω are the same as in Corollary 7.6. Proof. This is an immediate consequence of Corollary 7.6 and (15). We note that Kim and Oh [16] used Stanley’s combinatorial interpretation to study the Selberg integral. They found a connection with certain combinatorial objects called Young books, which 22 JANG SOO KIM AND DENNIS STANTON 0 0 0 1 1 1 2 3 3 0 0 1 2 2 2 4 0 0 1 1 1 1 6 6 0 2 2 8 8 4 8 Figure 6. The left diagrams is an element in RPP((4, 3, 1)) and the right diagram is an element in RPP((δ5+1 + (4, 3, 1))∗). generalizes standard Young tableaux of certain shapes. It would be interesting to see if their results can be generalized using ours. We also note that Kim and Okada [17] considered a different q- Selberg integral. By evaluating the q-integral, they showed that it is a generating function for Young books with certain weight. 8. Reverse plane partitions In this section we consider q-Selberg type integrals with Schur functions in the integrand. We then relate the resulting integral evaluations to generating functions for reverse plane partitions and generalized Gelfand-Tsetlin patterns. In particular we (1) define a Schur poset whose truncated order polytope has q-volume which is basically a Schur function sλ(x1, . . . , xn) (Lemma 8.6), (2) use q-integral evaluations to give generating functions for shifted and non-shifted reverse plane partitions with a given shape and diagonal (Theorem 8.7 and Corollary 8.11), (3) show that a q-integral of Warnaar may be interpreted as the trace generating function for reverse plane partitions of given shape (Theorems 8.12 and 8.13), and give an analogous Warnaar-type q-integral for the shifted version due to Gansner (Theorem 8.16), (4) define a generalized Gelfand-Tsetlin pattern, and give a new generating function for these objects (Theorem 8.10), (5) relate the Askey-Kadell-Selberg integral to a new weighted generating function for reverse plane partitions of square shape (Theorems 8.19 and 8.20). Before we get to the specific subsections, we need some notation and definitions. For a partition λ with `(λ) ≤ n, let (δn+1 + λ)∗ be the shifted Young diagram obtained from λ by attaching the shifted staircase δn+1 = (n, n − 1,..., 1). Definition 8.1. Let λ be a Young diagram or shifted Young diagram. A reverse plane partition of shape λ is a filling of λ with non-negative integers which are weakly increasing along rows and columns. We denote by RPP(λ) the set of reverse plane partitions of shape λ, see Figure 6. Definition 8.2. Let λ be a partition or a shifted partition. The k-diagonal of λ is the set of cells in λ which are in row i and column j with j − i = k. For T ∈ RPP(λ), the reverse diagonal of T is the sequence rdiag(T ) = (µ1, . . . , µn) obtained by reading the entries of T in the 0-diagonal of λ from southeast to northwest. Also the trace is defined by tr(T ) = µ1 + ··· + µn, and the sum of entries of T is denoted |T |. Example 8.3. Let T1 be the left diagram and T2 the right diagram in Figure 6. Then tr(T1) = 2, tr(T2) = 17, |T1| = 10, |T2| = 60, rdiag(T1) = (2, 0) and rdiag(T2) = (8, 8, 1, 0, 0). Note that the reverse diagonal of a reverse plane partition is always a partition. Definition 8.4. Let λ be a partition with `(λ) ≤ n. For 0 ≤ k ≤ λ1 + n − 1, let ak be the size (k) (k) of the k-diagonal of (δn+1 + λ)∗. We label the cells in the k-diagonal (δn+1 + λ)∗ by x1 , . . . , xak 0 from southeast to northwest. The 0-Schur poset PSchur(n, λ) is the poset on (k) {xi : 0 ≤ k ≤ λ1 + n − 1, 1 ≤ i ≤ ak} ON q-INTEGRALS OVER ORDER POLYTOPES 23 (0) x5 (1) x4 x(2) (0) 3 x4 (1) (3) x3 x3 (2) (4) x2 x2 x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8) (0) 5 4 3 3 2 2 2 1 1 x3 (1) (3) (5) (0) (1) (2) (3) (4) (5) (6) x2 x2 x2 x4 x3 x2 x2 x1 x1 x1 (2) (4) (6) x1 x1 x2 x(0) x(1) x(2) x(3) (0) 3 2 1 1 x2 (1) (3) (5) (7) (0) (1) x1 x1 x1 x1 x2 x1 (6) (8) x1 x1 x(0) (0) 1 x1 Figure 7. The left diagram shows the labels of the cells of (δn+1 + λ)∗, and the 0 right diagram shows the Hasse diagram of the 0-Schur poset PSchur(n, λ) for n = 5 and λ = (4, 3, 1). with relations x(k1) ≤ x(k2) if the cell x(k2) is weakly northwest of the cell x(k1) in (δ + λ) . See i1 i2 i2 i1 n+1 ∗ Figure 7. Then we define the Schur poset PSchur(n, λ) to be the poset obtained from the 0-Schur 0 (0) (0) poset PSchur(n, λ) by removing the elements x1 , . . . , xn . Note that we have ∼ 0 PSchur(n, λ) = PSchur(n − 1, λ) if `(λ) ≤ n − 1, ∼ 0 n PSchur(n, λ) = PSchur(n, λ − (1 )) if `(λ) = n. The next proposition shows that the Schur function sλ(x) can be written as a q-volume of a truncated order polytope of a Schur poset up to a constant factor. Proposition 8.5. Let λ be a partition with `(n) ≤ n and ak the size of the k-diagonal of (δn+1 + λ)∗. Let x = (x1, . . . , xn) be a sequence of fixed real numbers x1 < x2 < ··· < xn and ( (k) n (1) (2)+ λ {(xi ) ∈ R | | : xj ≤ xj ≤ xj+1, 1 ≤ j ≤ n − 1}, if `(λ) ≤ n − 1, I = (k) n (1) (2)+ λ {(xi ) ∈ R | | : xj 1 ≤ xj ≤ xj, 1 ≤ j ≤ n, x0 = 0}, if `(λ) = n, − (k) n (2)+ λ where (xi ) means a point in R | | with indices 1 ≤ k ≤ λ1 + n − 1 and 1 ≤ i ≤ ak. Then n Z Y (λ1+n 1) (λ1+n 2) (1) sλ(x)∆(x) = [λj + n − j]q! dqx − dqx − ··· dqx , (P (n,λ)) j=1 OI Schur (k) (k) (k) where dqx = dqx1 ··· dqxak . We will prove Lemma 8.6 which is equivalent to Proposition 8.5. The idea is to introduce new sets of variables which are interlacing with the previous sets of variables. Lemma 8.6. Let λ be a partition with `(n) ≤ n and ak the size of the k-diagonal of (δn+1 + λ)∗. (k) (k) (k) (0) For 0 ≤ k ≤ λ1 + n − 1, we consider sequences of variables x = (x1 , . . . , xak ) with xi = xi. Then for the set (λ +n 1) (0) Q = {x 1 − ≺ · · · ≺ x } of inequalities, we have n Z Y (λ1+n 1) (λ1+n 2) (1) (16) sλ(x)∆(x) = [λj + n − j]q! dqx − dqx − ··· dqx , j=1 Q (k) (k) (k) where dqx = dqx1 ··· dqxak . 24 JANG SOO KIM AND DENNIS STANTON λ1 Proof. We use induction on n. If n = 1, the left hand side of (16) is sλ(x1) = x1 . By Corollary 4.9, the right hand side of (16) is Z (λ1) (1) λ1 [λ1]q! dqx1 ··· dqx1 = x1 . 0 x(λ1) x(1) x ≤ 1 ≤···≤ 1 ≤ 1 Now suppose that (16) is true for n − 1 and consider n. CASE 1: `(λ) ≤ n − 1. By Lemma 5.7, we have n 1 Y− Z (17) sλ(x)∆(x) = [λj + n − j]q sλ(w)∆(w)dqw1 ··· dqwn 1. w x − j=1 ≺ Since `(λ) ≤ n − 1 and w = (w1, . . . , wn 1) has n − 1 variables, by the induction hypothesis, we have − n 1 − Z Y (λ1+n 2) (λ1+n 3) (1) (18) sλ(w)∆(w) = [λj + n − j − 1]q! dqw − dqw − ··· dqw , 0 j=1 Q (k) (k) (k) (0) where w = (w1 , . . . , w 0 ) is a sequence of variables for 0 ≤ k ≤ λ1 + n − 2 with w = wi, a0 ak i k is the size of the k-diagonal of (δn + λ)∗, and (λ +n 2) (0) Q0 = {w 1 − ≺ · · · ≺ w }. (k) (k+1) Note that ak0 = ak+1. By substituting wi = xi , we have Z Z (λ1+n 2) (λ1+n 3) (1) (19) dqw − dqw − ··· dqw dqw1 ··· dqwn 1 w x Q0 − ≺ Z (λ1+n 1) (λ1+n 2) (1) = dqx − dqx − ··· dqx . Q By (17), (18), and (19), we have n 1 − Z Y (λ1+n 1) (λ1+n 2) (1) sλ(x)∆(x) = [λj + n − j]q! dqx − dqx − ··· dqx . j=1 Q Since λn = 0, we have n 1 n Y− Y [λj + n − j]q! = [λj + n − j]q!, j=1 j=1 which implies that (16) is also true for n. (k) (k) (k) CASE 2: `(λ) = n. Let λn = m. Note that for 0 ≤ k ≤ m, we have x = (x1 , . . . , xn ). By Lemma 5.8, for 0 ≤ k ≤ m − 1, we have n Z (k) (k) Y (k+1) (k+1) (k+1) (20) sλ (kn)(x )∆(x ) = [λj −k+n−j]q sλ ((k+1)n)(x )∆(x )dqx . − x(k+1) x(k) − j=1 ≺ Applying (20) for k = 0, 1, . . . , m − 1, iteratively, we get m 1 n Z Y− Y (m) (m) (m) (1) (21) sλ(x)∆(x) = [λj − k + n − j]q sλ (mn)(x )∆(x )dqx ··· dqx . x(m) x(0) − k=0 j=1 ≺···≺ Since λ − (mn) has length at most n − 1, by CASE 1, we get n Z (m) (m) Y (λ1+n 1) (λ1+n 2) (m+1) (22) sλ (mn)(x )∆(x ) = [λj − m + n − j]q! dqx − dqx − ··· dqx , − 0 j=1 Q ON q-INTEGRALS OVER ORDER POLYTOPES 25 (λ1+n 1) (λ1+n 2) (m) where Q0 = {x − ≺ x − ≺ · · · ≺ x }. By (21) and (22), we have that sλ(x)∆(x) is equal to m 1 n n − Z Y Y Y (λ1+n 1) (λ1+n 2) (1) [λj − k + n − j]q [λj − m + n − j]q! dqx − dqx − ··· dqx , k=0 j=1 j=1 Q which is equal to the right hand side of (16). Thus (16) is also true for n. By induction (16) is true for all nonnegative integers n. 8.1. Shifted reverse plane partitions with fixed diagonal entries. The main result in this subsection, Theorem 8.7, reinterprets the q-integral evaluation in Lemma 8.6 as a generating function for shifted reverse plane partitions. Theorem 8.7. For λ, µ ∈ Parn we have b(δn+1+λ) X T q− µ+δn µ+δn µ+δn q| | = Qn q| |sλ(q )∆(q ). ∗ j=1(q; q)λj +n j T RPP((δn+1+λ) ) − ∈ rdiag(T )=µ Proof. Let µ1+n 1 µ2+n 2 µn µ+δn x = (x1, x2, . . . , xn) = (q − , q − , . . . , q ) = q . Following the notations in Proposition 8.5, we have n Z Y (λ1+n 1) (λ1+n 2) (1) sλ(x)∆(x) = [λj + n − j]q! dqx − dqx − ··· dqx . (P (n,λ)) j=1 OI Schur (k) (k) Let ω be the labeling of PSchur(n, λ) defined by ω(xi ) = j if dqxi is in the jth position of (λ1+n 1) (λ1+n 2) (1) dqx − dqx − ··· dqx . Then by Theorem 4.1, we have Z n (λ1+n 1) (λ1+n 2) (1) ( )+ λ X σ dqx − dqx − ··· dqx = (1 − q) 2 | | q| |, (P (n,λ)) OI Schur σ where the sum is over all (PSchur(n, λ), ω)-partitions σ such that (1) µj + n − j > σ(xj ) ≥ µj+1 + n − j − 1, 1 ≤ j ≤ n − 1 if `(λ) ≤ n − 1, (1) µj 1 + n − j + 1 > σ(xj ) ≥ µj + n − j, 1 ≤ j ≤ n if `(λ) = n, − where µ0 = ∞. There is an obvious way to regard such σ as a reverse plane partition U ∈ µ+δn RPP((δn+1 + λ)∗) with |σ| = |U| − |µ + δn| such that the rdiag(U) = q and in each column the entries are strictly increasing from top to bottom. Thus, by Theorem 4.1, we have Z n Pn (λ1+n 1) (λ1+n 2) (1) λ +( ) X U (µi+n i) dqx − dqx − ··· dqx = (1 − q)| | 2 q| |− i=1 − , (P (n,λ)) OI Schur U where the sum is over all column-strict reverse plane partitions U ∈ RPP((δn+1 + λ)∗) with rdiag(U) = qµ+δn . For such U, let T be the reverse plane partition obtained from it by decreasing the entries in row i by i − 1 for i = 1, 2, . . . , n. Then T ∈ RPP((δn+1 + λ)∗), rdiag(T ) = µ, and |U| = |T | + b(δn+1 + λ). Summarizing these, we obtain n n Pn Y λ +( ) X T +b(δn+1+λ) (µi+n i) sλ(x)∆(x) = [λj + n − j]q!(1 − q)| | 2 q| | − i=1 − , ∗ j=1 T RPP((δn+1+λ) ) ∈ rdiag(T )=µ which is equivalent to the theorem. 26 JANG SOO KIM AND DENNIS STANTON 8.2. Generalized Gelfand-Tsetlin patterns. In this subsection we restate Theorem 8.7 in terms of generalized Gelfand-Tsetlin patterns, see Theorem 8.10. Definition 8.8. For λ, µ ∈ Parn, an (n, λ, µ)-Gelfand-Tsetlin pattern is a triangular array {(xi,j) : 1 ≤ j ≤ n, 1 − λj ≤ i ≤ n + 1 − j} of nonnegative integers such that xi,j+1 ≥ xi,j ≥ xi+1,j and xi,n+1 i = µi. We assume that the − inequality holds if any index lies outside the stated domain. We denote by GTn(λ, µ) the set of (n, λ, µ)-Gelfand-Tsetlin patterns. Example 8.9. Let n = 4, λ = (3, 1) and µ = (3, 2, 2, 1). Then the coordinates of an (n, λ, µ)- Gelfand-Tsetlin pattern are arranged as follows. x1,4 x1,3 x2,3 x0,2 x1,2 x2,2 x3,2 x 2,1 x 1,1 x0,1 x1,1 x2,1 x3,1 x4,1 − − An example of an (n, λ, µ)-Gelfand-Tsetlin is shown below. 3 3 2 4 3 2 2 7 5 3 3 2 1 1 Note that if λ = ∅, then GTn(∅, µ) is basically the set of Gelfand-Tsetlin patterns, see [21, (7.37), p. 313]. There is a well known bijection [21, p. 314] between Gelfand-Tsetlin patterns and column strict tableaux which gives X T µ δn (23) q| | = q| |sµ(q ). T GT ( ,µ) ∈ n ∅ Note also that, if rotated by 180◦, the (λ, µ)-Gelfand-Tsetlin patterns are the same as the reverse plane partitions of shape (δn+1 + λ)∗ with reverse diagonal entries given by µ. Thus Theorem 8.7 can be restated as Theorem 8.10, which generalizes (23) to arbitrary λ. Theorem 8.10. For λ, µ ∈ Parn, we have n X T µ b(λ) δn µ+δn Y (q; q)n j q| | = q| |− sµ(q )sλ(q ) − . (q; q)λj +n j T GTn(λ,µ) j=1 − ∈ Equivalently, µ b(δn+λ) X T q| |− µ+δn µ+δn q| | = Qn sλ(q )∆(q ). j=1(q; q)λj +n j T GTn(λ,µ) − ∈ Proof. By Theorem 8.7, we have b(δn+1+λ) X T X T q− µ+δn µ+δn µ+δn q| | = q| | = Qn q| |sλ(q )∆(q ). ∗ j=1(q; q)λj +n j T GTn(λ,µ) T RPP((δn+1+λ) ) − ∈ ∈ rdiag(T )=µ Then we obtain the theorem using the fact [21, (7.105)] µj +n j − µi+n i µ+δn δn Y q − q − ∆(q ) sµ(q ) = i 1 j 1 = n 1 . q − q b(δn) Q − 1 i 8.3. The trace-generating function for reverse plane partitions. In this subsection we give two results on reverse plane partitions. In Corollary 8.11 we give a generating function for reverse plane partitions with a fixed shape and diagonal. Then we reinterpret an integral of Warnaar as the trace generating function for reverse plane partitions of a given shape, Theorem 8.13 and Corollary 8.14. The first result gives the generating function for reverse plane partitions with a fixed shape and diagonal by decomposing via the Durfee square. Here, the Durfee square of a partition λ is the largest square containing the cell in row 1 and column 1 in λ. n Corollary 8.11. Let λ = (λ1, . . . , λn), µ = (µ1, . . . , µn) and ν the partition obtained from (n ) by attaching λ to the right and µ0 at the bottom. Then for a partition ρ = (ρ1, . . . , ρn), we have n b(δn+1+λ) b(δn+1+µ) ρ+δ (2)− − n X T q q| | ρ+δn ρ+δn ρ+δn 2 (24) q| | = Qn sλ(q )sµ(q )∆(q ) . (q; q)λ +n j(q; q)µ +n j T RPP(ν) j=1 j − j − rdiag(∈ T )=ρ Proof. Since X T ρ X T X T q| | = q−| | q| | q| |, ∗ ∗ T RPP(ν) T RPP((δn+1+λ) ) T RPP((δn+1+µ) ) rdiag(∈ T )=ρ ∈ rdiag(T )=ρ ∈ rdiag(T )=ρ we get the desired formula by Theorem 8.10. The following q-integral evaluation is the special case k = 1 of Warnaar’s integral [24, Theo- rem 1.4], which has Macdonald polynomials instead of Schur functions. Theorem 8.12. [24, Special case of Theorem 1.4] Let λ = (λ1, . . . , λn) and µ = (µ1, . . . , µn) be partitions, Re(α) > −λn − µn and 0 < q < 1. Then Z n Y n n α 1 2 n α(2)+2(3) sλ(x)sµ(x) xi − ∆(x) dqx = (1 − q) q 0 x x 1 ≤ 1≤···≤ n≤ i=1 n 1 n n 1 n 1 Y− 2 Y 1 × sλ(1, q, . . . , q − )sµ(1, q, . . . , q − ) (q; q)i . 1 − qα+2n i j+λi+µj i=1 i,j=1 − − Proof. If k = 1 in [24, Theorem 1.4], we have Z n Y α 1 Y 2 xi sλ(x)sµ(x) xi − xj dqx n xj [0,1] i=1 1 i By the same argument as in the proof of (15), one can show that these two integral evaluations are equivalent. We now show that Warnaar’s q-integral Theorem 8.12 can be interpreted as a trace-generating function of a reverse plane partition of given shape. First we show that the trace generating function is a q-integral. 28 JANG SOO KIM AND DENNIS STANTON n Theorem 8.13. Let λ = (λ1, . . . , λn), µ = (µ1, . . . , µn) and ν the partition obtained from (n ) by attaching λ to the right and µ0 at the bottom. Then n (1 a)(2) b(δn+1+λ) b(δn+1+µ) X T a tr(T ) q − − − q| |(q ) = Qn (q; q)λ +n j(q; q)µ +n j T RPP(ν) j=1 j − j − ∈ n 1 Z Y × a 2 n sλ(x)sµ(x) xi ∆(x) dqx. (1 − q) 0 x x 1 ≤ 1≤···≤ n≤ i=1 Proof. Let ρ = (ρ1, . . . , ρn) be a fixed partition. For T ∈ RPP(ν) with rdiag(T ) = ρ, we have n a a tr(T ) a ρ a( ) ρ+δn (25) q · = q | | = q− 2 q| | . Thus, by Corollary 8.11, n a+1 (1 a)(2) b(δn+1+λ) b(δn+1+µ) ρ+δn X T a tr(T ) q − − − q| | (26) q| |(q ) = Qn (q; q)λ +n j(q; q)µ +n j T RPP(ν) j=1 j − j − rdiag(∈ T )=ρ ρ+δn ρ+δn ρ+δn 2 × sλ(q )sµ(q )∆(q ) . By Lemma 4.3, if we take the sum of both sides of (26) over all ρ ∈ Parn, we get the theorem. As a corollary of Theorems 8.12 and 8.13, one can prove the trace-generating function for reverse plane partitions, which is a special case of [9, Theorem 5.1]. One uses the principally specialized Schur functions as products, [21, Lemma 7.21.1] and [21, Lemma 7.21.2]. We do not give the details. Corollary 8.14. Let ν be a partition. Then X a tr(T ) T Y 1 (q ) q| | = , 1 − (qa)χ(u)qh(u) T RPP(ν) u ν ∈ ∈ where χ(u) = 1 if u is in the Durfee square of ν and χ(u) = 0 otherwise. 8.4. The trace-generating function for shifted reverse plane partitions. In this subsection we find a “shifted”-counterpart to Warnaar’s q-integral, which gives a trace generating function for shifted reverse plane partitions. We need a special case of Gansner’s result on the trace-generating function for shifted reverse plane partitions. We note that Gansner [9] considered a more general weight which involves the entries in the k-diagonal for each k. Theorem 8.15. [9, A special case of Theorem 7.1] For λ ∈ Parn, we have X tr(T ) T Y 1 x q| | = , 1 − xχ(u)qh(u) T RPP((λ+δ )∗) u (λ+δ )∗ ∈ n+1 ∈ n+1 where χ(u) = 1 if u is in column j for 1 ≤ j ≤ n, and χ(u) = 0 otherwise, and h(u) is the length of the shifted hook at u, see [9] for the definition. Equivalently, this can be restated as n X tr(T ) T b(λ) n 1 Y (q; q)j 1 x q| | = q− sλ(1, q, . . . , q − ) − ∗ (q; q)λj +n j T RPP((λ+δn+1) ) j=1 − ∈ Y 1 × . 1 − xq1+2n i j+λi+λj+1 1 i j n − − ≤ ≤ ≤ We do not prove the equivalence of the second statement. It follows from routine facts on Schur functions. Theorem 8.15 gives the generating function for shifted reverse plane partitions, but we want to find the corresponding q-integral. Here is the “shifted” version of Warnaar’s q-integral which corresponds to Gansner’s theorem. ON q-INTEGRALS OVER ORDER POLYTOPES 29 0 1 3 0 3 3 2 3 3 Figure 8. A reverse plane partition of square shape (33). Theorem 8.16. If λ ∈ Parn, Z n Y n n α 1 n α(2)+(3) n 1 sλ(x) xi − ∆(x)dqx = (1 − q) q sλ(1, q, . . . , q − ) 0 x x 1 ≤ 1≤···≤ n≤ i=1 n 1 Y− Y 1 × (q; q)i . 1 − qα+2n i j+λi+λj+1 i=1 1 i j n − − ≤ ≤ ≤ Proof. Let a = α − 1. By Theorem 8.10, we have b(δ +λ) n q n+1 X a tr(T ) T a( ) a µ+δn − µ+δn µ+δn µ+δn (q ) q| | = q− 2 q | | Qn q| |sλ(q )∆(q ). ∗ j=1(q; q)λj +n j T RPP((δn+1+λ) ) − ∈ rdiag(T )=µ By summing both sides over all µ ∈ Parn and using Lemma 4.3, we obtain a n b(δ +λ) − (2)− n+1 X a tr(T ) T q X µ+δn a µ+δn µ+δn µ+δn (q ) q| | = Qn q| |q | |sλ(q )∆(q ) ∗ j=1(q; q)λj +n j T RPP((δn+1+λ) ) − µ Parn ∈ ∈ (a+1)(n) (n) b(λ) Z n q− 2 − 3 − 1 Y = s (x) xa∆(x)d x. Qn − n λ i q j=1(q; q)λj +n j (1 q) 0 x1 xn 1 − ≤ ≤···≤ ≤ i=1 We finish the proof by comparing this with Theorem 8.15. 8.5. Reverse plane partitions and the q-Selberg integral. In this subsection we show that the Askey-Kadell-Selberg integral is equivalent to a generating function for reverse plane partitions of a square shape with certain weight. The weight has 3 parameters and generalizes the sum of the entries of a reverse plane partition. n Definition 8.17. For a reverse plane partition T ∈ RPP(n ) with rdiag(T ) = (µ1, . . . , µn), let vi = µi + n − i for i = 1, 2, . . . , n. For integers a, b ≥ 0 and m ≥ 1, we define vj 2m 1 1 m+vi vj T +a tr(T ) v1+1 vn+1 Y (q ) − (q − − ; q)2m 1 wta,b,m(T ) = q| | · (q , . . . , q ; q)b − . qvj − qvi 1 i T +a tr(T ) v1+1 vn+1 wta,b,1(T ) = q| | · (q , . . . , q ; q)b, T +a tr(T ) wta,0,1(T ) = q| | · , T wt0,0,1(T ) = q| |. 30 JANG SOO KIM AND DENNIS STANTON The following theorem shows that the Askey-Kadell-Selberg integral is a generating function for reverse plane partitions of a square shape, up to simple constants. Theorem 8.19. The weighted generating function for reverse plane partitions of a square shape is given by the Askey-Kadell-Selberg integral ( 1 a)(n) 2(n) X q − − 2 − 3 wta,b,m(T ) = n 1 · AKSn(a + 1, b + 1, m). − n Q − 2 T RPP(nn) (1 q) j=1 (q; q)j ∈ Proof. Let us fix µ = (µ1, . . . , µn) ∈ Parn and v = (v1, . . . , vn) with vi = µi + n − i. Then X X T +a tr(T ) µ+δn wta,b,m(T ) = q| | · C(q ), T RPP(nn) T RPP(nn) rdiag(∈ T )=µ rdiag(∈ T )=µ where n Y Y 2m 1 1 m 1 C(x) = (qxi; q)b xj − (q − xi/xj; q)2m 1∆(x)− . − i=1 1 i n (1 a) 2b(δn+1) µ+δ a+1 − (2)− n X T +a tr(T ) q (q| |) µ+δn 2 q| | · = n 1 ∆(q ) . Q − 2 T RPP(nn) j=1 (q; q)j rdiag(∈ T )=µ Combining these two equations we get n (1 a) 2b(δn+1) µ+δ a+1 − (2)− n X q (q| |) µ+δn 2 µ+δn wta,b,m(T ) = n 1 ∆(q ) C(q ). Q − 2 T RPP(nn) j=1 (q; q)j rdiag(∈ T )=µ By summing over all µ ∈ Parn and using Lemma 4.3, we get n (1 a)( ) 2b(δn+1) X q − 2 − 1 wta,b,m(T ) = n 1 n AKSn(a + 1, b + 1, m). Q − 2 (1 − q) T RPP(nn) j=1 (q; q)j ∈ n n Then we finish the proof using the fact b(δn+1) = 2 + 3 . Using Theorem 8.19 and the evaluation (15), we obtain a product formula for the weighted generating function. Theorem 8.20. The weighted generating function for reverse plane partitions of square shape has the explicit product formula (1 a+m+am)(n) 2(n+1)+2m2(n) X q − 2 − 3 3 wt (T ) = a,b,m (1 − q)n2 T RPP(nn) ∈ n Y [a + (j − 1)m]q![b + (j − 1)m]q![jm − 1]q! × . [a + b + (n + j − 2)m + 1] ![m − 1] !([j − 1] !)2 j=1 q q q We conclude with some special cases of Theorem 8.20. If m = 1 in Theorem 8.20, we have Corollary 8.21. A weighted generating function for reverse plane partitions of square shape is n − − X T +a tr(T ) v1+1 vn+1 1 Y [a + j 1]q![b + j 1]q! q| | · (q , . . . , q ; q)b = n2 . (1 − q) [a + b + n + j − 1]q![j − 1]q! T RPP(nn) j=1 ∈ where the reverse diagonal entries of T are (µ1, . . . , µn), and vi = µi + n − i, 1 ≤ i ≤ n. ON q-INTEGRALS OVER ORDER POLYTOPES 31 If b = 0 in Corollary 8.21, we have the trace generating function for square shapes, see [21]. If also a = 0, we have the generating function for reverse plane partitions of square shape. Kamioka [15] has another weighted generating function for reverse plane partitions of a square shape. 9. q-Ehrhart polynomials In this section, using q-integrals, we study q-Ehrhart polynomials and q-Ehrhart series of order polytopes with some faces removed. We refer the reader to [7] for the more details in q-Ehrhart polynomials of lattice polytopes. This section was initiated by discussions with Josuat-Verg`es. Throughout this section we assume that P is a poset on {x1, . . . , xn} and ωn is the bijection from {x1, . . . , xn} to [n] given by ωn(xi) = i for 1 ≤ i ≤ n. We denote by P the dual of P , i.e., the poset obtained by reversing the orders in P . Definition 9.1. For a bounded set X of points in Rn and a positive integer m, we define the q-Ehrhart function X x1+ +xn Eq(X, m) = q ··· . (x ,...,x ) mX n 1 n ∈ ∩Z Definition 9.2. Let P be a poset on {x1, . . . , xn}. We define ∆(P ) to be the set of points n (x1, . . . , xn) ∈ [0, 1] such that • xi ≥ xj if xi ≤P xj, • xi > xj if xi ≤P xj and i > j. Note that ∆(P ) is obtained from the order polytope O(P ) by removing some faces. In general ∆(P ) is not a polytope. However, if (P, ωn) is naturally labeled, then ∆(P ) = O(P ). Therefore, by adding the assumption that (P, ωn) is naturally labeled, every result in this section has a corollary stated in terms of the order polytope O(P ). For instance, see Corollary 9.4. Now we show some properties of the q-Ehrhart function of ∆(P ): it is represented as a q-integral of a truncated order polytope of P and it is a polynomial in [m]q whose leading coefficient is the q-volume of the order polytope of P . Theorem 9.3. Let P be a poset on {x1, . . . , xn}. Then, for an integer m ≥ 0, we have 1 E (∆(P ), m) = V (O m+1 n (P )), q (1 − q)n q [q ,1] and, equivalently, X m + n − des(π) E (∆(P ), m) = qmaj(π) . q n π (P,ω ) q ∈L n Moreover, the q-Ehrhart function Eq(∆(P ), m) is a polynomial in [m]q whose coefficients are rational functions in q and whose leading coefficient is the q-volume Vq(O(P )). Proof. By definition, we have X σ Eq(∆(P ), m) = q| |, σ where the sum is over all (P, ωn)-partitions σ with max(σ) ≤ m. By Theorem 4.1, this is equal to 1 (1 q)n Vq(O[qm+1,1]n (P )). −By Corollary 4.2, we have X Vq(O[qm+1,1]n (P )) = Vq(O[qm+1,1]n (Pπ)). π (P,ω ) ∈L n By Corollary 4.6, we have n maj(π) m + n − des(π) Vq(O[qm+1,1]n (Pπ)) = (1 − q) q , n q 32 JANG SOO KIM AND DENNIS STANTON which is a polynomial in [m]q whose coefficients are rational functions of q. Since Eq(∆(P ), m) is a n sum of these q-volumes divided by (1 − q) , it is also a polynomial in [m]q. The leading coefficient of Eq(∆(P ), m) as a polynomial in [m]q is Eq(∆(P ), m) 1 lim = lim Vq(O[qm+1,1]n (P )) = Vq(O(P )). m n m − m n →∞ [m]q →∞ (1 q ) If we add the assumption that (P, ωn) is naturally labeled in Theorem 9.3 then we obtain a result on the order polytope O(P ). Corollary 9.4. Let P be a poset on {x1, . . . , xn}. Suppose that (P, ωn) is naturally labeled. Then, for an integer m ≥ 0, we have 1 E (O(P ), m) = V (O m+1 n (P )). q (1 − q)n q [q ,1] Moreover, the q-Ehrhart function Eq(O(P ), m) is a polynomial in [m]q whose coefficients are rational functions in q and whose leading coefficient is the q-volume Vq(O(P )). We note that Eq(X, m) is not always a polynomial in [m]q. For example, if n = 1 and X = {1/k} m/k then Eq(X, m) = q if m is divisible by k and Eq(X, m) = 0 otherwise. Remark 9.5. In [7], Chapoton defines the q-volume of a poset P on {x1, . . . , xn} to be the leading coefficient of the q-Ehrhart polynomial Eq(O(P ), m) times [n]q!. If (P , ωn) is naturally labeled, we have ∆(P ) = O(P ). By Corollary 9.4, the leading coefficient of the q-Ehrhart polynomial Eq(O(P ), m) is the q-volume Vq(O(P )). Thus, Chapoton’s q-volume of P is equal to [n]q!Vq(O(P )). Next we consider the q-Ehrhart series. Definition 9.6. For a set X of points in Rn, we define the q-Ehrhart series of X by X m Eq∗(X, t) = Eq(X, m)t . m 0 ≥ We now show that the q-Ehrhart series of ∆(P ) is a generating function for the linear extensions of P . Corollary 9.7. For a poset P on {x1, . . . , xn}, we have 1 X des(π) maj(π) Eq∗(∆(P ), t) = t q . (t; q)n+1 π (P,ω ) ∈L n Proof. By Theorem 9.3, we have X m X maj(π) m + n − des(π) E∗(∆(P ), t) = t q . q n m 0 π (P,ω ) q ≥ ∈L n Thus, it suffices to show that X m + n − des(π) tdes(π) (27) tm = . n (t; q)n+1 m 0 q ≥ The summand in the left hand side of (27) is zero unless m ≥ des(π). By shifting the index m by m + des(π), the left hand side of (27) becomes X m + n 1 tdes(π) tm = tdes(π) · , n (t; q)n+1 m 0 q ≥ where the q-binomial theorem is used. This completes the proof. ON q-INTEGRALS OVER ORDER POLYTOPES 33 Let us consider the special case of Corollary 9.7 when P is the anti-chain on {x1, . . . , xn}. In n n m n this case we have ∆(P ) = [0, 1] and L(P, ωn) = Sn. Since Eq([0, 1] , m) = (1 + q + ··· + q ) , Corollary 9.7 reduces to MacMahon’s identity X X tdes(π)qmaj(π) = tm(1 + q + ··· + qm)n. π S m 0 ∈ n ≥ We note that Beck and Braun [4] also gave a proof this identity by decomposing [0, 1]n into simplices. We finish this section by showing that the q-Ehrhart series of ∆(P ) has a q-integral represen- tation. Note that if 0 < t < 1 and 0 < q < 1, then logq t > 0. Corollary 9.8. Let 0 < t < 1 and 0 < q < 1. For a poset P on {x1, . . . , xn}, we have 1 Z log t 1 q − ··· Eq∗(∆(P ), t) = n+1 x0 dqx0dqx1 dqxn. (1 − q) (P 0) O where P 0 is the poset obtained from P by adding a new element x0 which is smaller than all elements in P . Proof. By Theorems 9.3 and 4.1, X m 1 X m X σ E∗(∆(P ), t) = t · V (O m+1 n (P )) = t q| |, q (1 − q)n q [q ,1] m 0 m 0 σ ≥ ≥ where the second sum in the rightmost side is over all (P, ωn)-partitions σ with max(σ) ≤ m. This can be rewritten as X σ0(x ) σ0 σ0(x ) X σ0(x ) log t 1 σ0 t 0 q| |− 0 = (q 0 ) q − q| |, σ0 σ0 where the sum is over all (P 0, ω0)-partitions σ0 for the labeling ω0 of P 0 given by ω0(xi) = i for 0 ≤ i ≤ n. By applying Theorem 4.1 again, we obtain the desired identity. Acknowledgements The authors are grateful to Matthieu Josuat-Verg`esfor helpful discussion. They also thank Ira Gessel for pointing out MacMahon’s result in Theorem 4.8. References [1] G. E. Andrews and R. Askey. Another q-extension of the beta function. Proc. Amer. Math. Soc., 81(1):97–100, 1981. [2] G. E. Andrews, R. Askey, and R. Roy. 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(Jang Soo Kim) Department of Mathematics, Sungkyunkwan University, Suwon, South Korea E-mail address: [email protected] (Dennis Stanton) School of Mathematics, University of Minnesota, Minneapolis, MN 55455 E-mail address: [email protected]