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15 A 1 3 4 2 14 9 10 8 13 12 11 7 5 6
Figure 1. A Laguerre configuration (A, f) on [15] with A = [15] 3, 8, 9, 11 . \{ } and is the linear functional defined by L 1 ∞ α x (f)= f(x)x e− dx. (1.4) L Γ(α + 1) Z0 The linearization formula [23] reads as follows: n ! n ! n !2n1+n2+n3 2s (α + 1) (L(α)(x)L(α)(x)L(α)(x)) = 1 2 3 − s . (1.5) L n1 n2 n3 (s n )!(s n )!(s n )!(n + n + n 2s)! s 0 1 2 3 1 2 3 X≥ − − − − A combinatorial model for Laguerre polynomials with parameter α was first given by Foata and Strehl [7]. Recall that a Laguerre configuration on [n] := 1,...,n is a pair (A, f), where A [n] and f is an injection from A to [n]. A Laguerre{ configuration} can be depicted by a⊂ digraph on [n] by drawing an edge i j if and only if f(i)= j. Clearly such a graph has two types of connected components called→ cycles and paths, see Figure 1. Let n,k be the set of Laguerre configurations (A, f) on [n] with A = n k. Then Foata andLC Strehl’s interpretation [7] reads | | − n! n + α (α + 1)cyc(f) = , (1.6) k! n k (A,f) n,k X∈LC − where cyc(f) is the number of cycles of f. Note that one can derive (1.6) from any of the three formulae (1.1)–(1.3), see [1, 7]. The aim of this paper is to study combinatorial aspects of more general (q,y)-Laguerre (α) polynomials Ln (x; y; q) (n 0) defined by the three term-recurrence relation ≥ L(α) (x; y; q)=(x (y[n + α + 1] + [n] )) L(α)(x; y; q) (1.7) n+1 − q q n (α) y[n]q[n + α]q Ln 1(x; y; q) (α 1, n 1) − − ≥ − ≥ (α) (α) with L0 (x; y; q) = 1, L 1 (x; y; q) = 0. Here and throughout this paper, we use the − 1 qn n standard q-notations: [n] = − for n 0, the q-analogue of n-factorial n! = [i] q 1 q ≥ q i=1 q and the q-binomial coefficient − Q n n! = q for 0 k n. k k! (n k)! ≤ ≤ q q − q S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 3
(α) (α) Clearly we have Ln (x;1;1) = Ln (x). Kasraoui et al. [17] gave a combinatorial inter- (0) pretation for the linearization coefficients of the polynomials Ln (x; y; q) and pointed out (0) that a combinatorial model for Ln (x; y; q) can be derived from Simion and Stanton’s model for octabasic q-Laguerre polynomials in [22]. For k Z let ∈ N := n Z : n k k { ∈ ≥ } and N := N1. Recently, using the theory of q-Riordan matrices Cheon, Jung and Kim [3] (α) derived a combinatorial model for the q-Laguerre polynomials Ln (x; q; q) when α N0. It is then natural to seek for a combinatorial structure unifying the above two special∈ cases, as was alluded at the end of [3]. Our first goal is to give such a combinatorial (α) model for Ln (x; y; q) with variable y and integer α N 1 by using a q-analogue of ∈ − Foata–Strehl’s Laguerre configurations. Moreover, when α N0 the (q,y)-Laguerre poly- (α) ∈ nomials Ln (x; y; q) are orthogonal polynomials, we use the combinatorial theory of con- tinued fractions to give a combinatorial interpretation for the moments of (q,y)-Laguerre polynomials and prove that the linearization coefficients are polynomials in y and q with nonnegative integral coefficients. (α) By (1.7) the first few values of Ln (x; y; q) are as follows: L(α)(x; y; q)= x y[α + 1] , 1 − q L(α)(x; y; q)= x2 (y[α + 1] + y[α + 2] + 1) x + [α + 1] [α + 2] y2, 2 − q q q q L(α)(x; y; q)= x3 y([α + 1] + [α + 2] + [α + 3] )+2+ q)x2 3 − q q q 2 + y ([α + 1]q[α + 2]q + [α + 2]q[α + 3]q + [α + 1]q[α + 3]q) + y([α + 3] + [2] [α + 1] ) + [2] ) x y3[α + 1] [α + 2] [α + 3] . q q q q − q q q For convenience, we introduce the signless (q,y)-Laguerre polynomials n L(α)(x; y q) := ( 1)nL(α)( x; y; q)= ℓ(α)(y; q)xk. (1.8) n | − n − n,k Xk=0 N (α) For α 1 we observe that ℓn,k(y; q) are polynomials in y, q with nonnegative integral ∈ − coefficients, which is far from obvious from the explicit formula (2.8). For α N 1, (α) ∈ − formula (1.6) implies that ℓn,k(1; 1) is equal to the number of Laguerre configurations in such that each cycle carries a color [1 + α]. In particular, the number of Laguerre LCn,k ∈ configurations in n,k without cycles (i.e., consisting of only k paths) is equal to the Lah number [18]: LC ( 1) n! n 1 ℓ − (1;1) = − . n,k k! k 1 − Remark 1. Two different q-analogues of Lah numbers were defined and studied by Garsia and Remmel [9] and Lindsay et al. [19], respectively. Moreover an elliptic analogue of Garsia and Remmel’s q-Lah numbers was constructed by Schlosser and Yoo [21]. 4 QIONGQIONGPANANDJIANGZENG
The organization of this paper is as follows. In Section 2 we identify the(q,y)-Laguerre polynomials as a rescaled version of Al-Salam–Chihara polynomials and derive several ex- pansion formulae for (q,y)-Laguerre polynomials. In Section 3 we present a combinatorial interpretation for the (q,y)-Laguerre polynomials in terms of α-Laguerre configurations, which are in essence the product structure of ”cycles” and ”paths”. In Section 4 we give a combinatorial interpretation for the moments of (q,y)-Laguerre polynomials and prove that the linearization coefficients are polynomials in y and q with nonnegative in- tegral coefficients. As the Laguerre polynomials play an important role in the theory of rook polynomials, we translate our α-Laguerre configurations in terms of rook placements in Section 5 and set up the connection between our α-Laguerre configurations and the matching model of complete bipartite graphs Kn,n+α (see Godsil and Gutman [12]).
2. A detour to Al-Salam–Chihara polynomials
The q-Pochhammer symbol or q-shifted facorial (a; q)n is defined by
n 1 i + − (1 aq ) for n Z , (a; q) = i=0 − ∈ ∪ {∞} n 1 for n =0. (Q
The Al-Salam–Chihara polynomials Qn(x) := Qn(x; a, b q) are defined by the generating function [16, Chaper 14]: |
∞ tn (at, bt; q) Q (x; a, b q) = ∞ (2.1) n | (q; q) (teiθ, te iθ; q) n=0 n − X ∞ with (a, b; q) =(a; q) (b; q) , and satisfy the recurrence relation (op. cit.): ∞ ∞ ∞
Q0(x)=1, Q 1(x)=0, − n n n 1 (2.2) (Qn+1(x)=(2x (a + b)q )Qn(x) (1 q )(1 abq − )Qn 1(x), n 0. − − − − − ≥ We have the explicit formula
n n 1 (ab; q)n (q− ; q)k(au; q)k(au− ; q)k k Qn(x; a, b q)= n q , (2.3) | a (ab; q)k(q; q)k Xk=0 − u+u 1 iθ where x = 2 or x = cos θ if u = e . (α) Comparing (1.7) with (2.2) and using (1.8) we see that our polynomials Ln (x; y q) are a re-scaled version of the Al-Salam–Chihara polynomials: |
n (α) √y (1 q)x + y +1 1 α+1 L (x; y q)= Qn − ; , √yq q . (2.4) n | 1 q 2√y √y | − S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 5
The Al-Salam–Chihara polynomials (see [16, p. 455–456] and [14]) are orthogonal with respect to the linear functional ˆ : Lq (q, ab; q) +1 f(x)dx ∞ 1 2(2x2 1)qk + q2k ˆq(f)= ∞ − − . (2.5) 2 k 2 2k k 2 2k L 2π 1 √1 x [1 2xaq + a q ][1 2xbq + b q ] Z− − kY=0 − − (α) Hence, for α N0, the polynomials Ln (x; y q) are orthogonal with respect to the linear functional ∈: | Lq (q, qα+1; q) 1 q B+ f(x)dx q(f)= ∞ − 2 L 2π 2√y B− 1 v(x) Z − ∞ p [1 2(2v(x)2 1)qk + q2k] − − , (2.6) × [1 2v(x)qk/√y + q2k/y][1 2v(x)qk+α+1√y + q2k+2α+2y] kY=0 − − (1 √y)2 ± where B = 1 q and ± − 1 v(x)= ((q 1)x +(y + 1)). (2.7) 2√y − Now, by (2.4) we derive the explicit formula from (2.3)
n k 1 − (α) n!q n + α k(k n) n k j Ln (x; y q)= q − y − x + (1 yq− )[j]q , (2.8) | k!q k + α − k=0 q j=0 X Y and the generating function from (2.1) tn (α)(x; y; t q) := L(α)(x; y q) L | n | n! n 0 q X≥ (t; q) (ytqα+1; q) = ∞ ∞ , (2.9) ∞ [1 ((1 q)x + y + 1)tqk + yt2q2k] k=0 − − which can be written as Q (α) (α) ( 1) (x; y; t q)= (0; y; t q) − (x; y; t q). (2.10) L | L | ·L | Define the “vertical generating function” n (α) k (α) (α) t k (y; t q) := [x ] (x; y; t q)= ℓn,k(y, q) , (2.11) L | L | n!q n k X≥ and the q-derivative operator for f(t) R[[t]] by Dq ∈ f(t) f(qt) (f(t)) = − , Dq (1 q)t − n n 1 where R = C[[x, y, q, . . .]]. Thus (1) = 0 and (t ) = [n] t − for n> 0. Dq Dq q 6 QIONGQIONGPANANDJIANGZENG
It follows from (2.9) that
( 1) x ( 1) − (x; y; t q)= − (x; y; t q), (2.12) DqL | (1 t)(1 yt)L | − − which in particular gives
( 1) ( 1) − (y; t q) = [x] − (x; y; t q) DqL1 | DqL | 1 = (1 t)(1 ty) − − tn = n! [n + 1] . (2.13) q y n! n 0 q X≥ So we can rewrite (2.12) as
( 1) ( 1) ( 1) − (x; y; t q)= x − (y; t q) − (x; y; t q), (2.14) DqL | ·DqL1 | ·L | which is equivalent to the following result. Proposition 1. For n N we have ∈ n ( 1) n ( 1) Ln−+1 (x; y q)= x k!q[k + 1]y Ln− k (x; y q). (2.15) | k q − | Xk=0 Now, applying the q-binomial formula (see [10, Chapter 1])
(a; q)n n (az; q) z = ∞ (q; q) (z; q) n 0 n X≥ ∞ with a = qα+1 and z = yt we have
α+1 n n (α) (ytq ; q) (yt) (0; y; t q)= ∞ = [α + k]q . (2.16) L | (yt; q) n!q n 0 k=1 ∞ X≥ Y Plugging the latter into (2.10) gives the following result. Proposition 2. For n N we have ∈ n k (α) n k ( 1) Ln (x; y q)= [α + j]q y Ln− k (x; y q). (2.17) | k − | q j=1 Xk=0 Y Remark 2. (1) More generally we can prove the following connection formula for α β 1: ≥ ≥ − n k 1 − (α) n β+1 k (β) Ln (x; y q)= [α β + j]q (yq ) Ln k(x; y q). (2.18) | k − − | q j=0 Xk=0 Y S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 7
(2) If q 1 identity (2.9) reduces to → n x/(1 y) (α) t (α+1) (1 y)t − − L (x; y 1) = (1 yt)− 1 − . n | n! − − 1 yt n 0 X≥ − Comparing with the generating function of the Meixner polynomials [16, (1.9.11)]
∞ (β)n n x β x M (x; β,c)t = (1 t)− − (1 t/c) , n! n − − n=0 X we derive x L(α)(x; y 1) = yn(α + 1) M − ; α +1,y . n | n n 1 y − (α) Hence the (q,y)-Laguerre polynomials Ln (x; y q) are a q-analogue of rescaled Meixner polynomials. |
3. Combinatorial interpretation of (q,y)-Laguerre polynomials The reader is referred to [1, 6, 13] for the general combinatorial theory of exponential generating functions for labelled structures. For our purpose we need only a q-version of this theory for special labelled structures. A labelled structures on a (finite) set A N is ⊂ a graph with vertex set A. Consider a family of labelled -structures = n∞=0 n, where consists of the -structures on [n]. If A = a ,...,aF N, whereF a∪
We need some further definitions. (a) For any permutation σ of a set A N let the wordσ ˆ denote its linear rep- resentation in the usual sense, i.e.,σ ˆ⊂ = σ(i ) ...σ(i ) if A = i ,...,i with 1 n { 1 n} i1 < < in. (b) A list··· of (nonnegative) integers, taken as a word over N, is strict if no element occurs more than once. For a strict list ρ let rl(ρ) be the number of elements that come after the maximum element. (c) For a set λ of k non-empty and disjoint strict lists of integers, order these lists according to their minimum element (increasing). This gives a list of k words (λ1,...,λk), which will be identified with λ. Then λ = λ1 ...λk denotes the concatenation of these lists. Two particular structures will be used to interpret the (q,y)-Laguerre polynomials. (d) The structures (α) consist of permutations σ, where each cycle carries a color 0, 1, 2,...,α .S Write σ as a product of unicolored permutations, σ = σ σ σ∈, { } 0 · 1 ··· α where σi is the product of cycles with color i. Now consider the concatenation σ =σ ˆ σˆ σˆ 0 · 1 ··· α and the word on 0, 1 : { } σˆ0 σˆ1 σˆα σ =0| |10| |1 10| |. ··· Define the valuation u on (α) by S σ inv(σ)+inv(σ) u(σ)= y| |q . (k) (e) The structures in consist of sets λ =(λ1,...,λk) of k nonempty and disjoint strict lists (cf. (c)).L Define the valuation v on in(k) by L rl(λ) inv(λ) rl(λ) v(λ)= y q − , k where rl(λ)= i=1 rl(λi). Let (α) := (α) in(k)[n]. For any α-Laguerre configuration (σ, λ) (α)[A] LCn,k S P·L ∈ S × in(k)[B] with A B = and A B = [n], in order to invoke the folklore statement (3.1) Lone should use as∩ valuation∅ ∪ w(σ, λ)= u(λ) v(λ) qinv(A,B) · · σ inv(σ)+inv(σ) rl(λ) inv(λ) rl(λ) inv(A,B) = y| |q y q − q σ +rl(λ) inv(σ)+inv(σ)+inv(λ) rl(λ) inv(A,B) = y| | q − q σ +rl(λ) inv(σ.λ) rl(λ)+inv(σ) = y| | q − . (3.2) The essential point is inv(σ)+ inv(λ)+ inv(A, B)= inv(σ.λ). This describes the weighted configurations ( (α)) . An element of ( (α)) is called an α-Laguerre configuration on LCn,k w LCn,k w [n], see Figure 2. S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 9
15 σ λ 1 3 4 0 2 14 9 10 8 1 0 1 13 12 11 7 5 6
(1) Figure 2. A 1-Laguerre configuration (σ, λ) 15,4, which is the La- guerre configuration in Figure 1 of which each cycle∈ LC gets a color 0 or 1.
Lemma 3. For α N, we have ∈ (α)(t)= (α)(0; y; t q). Su L | Proof. Let P(n, α) be the set of words of length n+α with n 0’s and α 1’s, i.e., lattice paths from (0, 0) to (n, α). For σ (α)[n] the word σ can be seen as the linear representation of an (ordinary) permutation∈ Sσ ˜ (0)[n], whereas σ P(n, α). The mapping ∈ S ∈ (α)[n] (0)[n] P(n, α): σ (˜σ, σ) S → S × 7→ is a bijection, and from summing both contributions separately one obtains qinv(σ)+inv(σ) = qinv(σ) qinv(σ) σ (α)[n] σ (0)[n] σ P (n,α) ∈SX ∈SX ∈X n + α = n! , q α q n which is i=1[α + i]q. So we get n Q (α)(t)= [α + i] (yt)n. Su q n 0 i=1 X≥ Y The result then follows from (2.16). Lemma 4. For integer k 1, we have ≥ (k) ( 1) in (t)= − (y; t q). L v Lk | Proof. We proceed by induction on k 1. ≥ (1) The case k = 1. For a single list λ = λ in [n + 1], let jλ be the position of • (1)∈ L the maximum element, let λ′ = λ′ in [n] be the list obtained by deleting this maximum element. Then ∈L (1) (1) in [n + 1] in [n] [n + 1] : λ (λ′, j ) L →L × 7→ λ 10 QIONGQIONGPANANDJIANGZENG
is a bijection such that inv(λ)= inv(λ′)+ rl(λ), one gets rl(λ) inv(λ) rl(λ) inv(λ′) n+1 j y q − = q y − λ in(1)[n+1] λ′ in(1)[n] j [n+1] ∈L X ∈LX ∈X and thus v(λ)= n!q[n + 1]y, λ in(1)[n+1] ∈L X which, in view of (2.13), gives (1) ( 1) in (t)= − (y; t q), DqL v DqL1 | and by q-integration (1) ( 1) in (t)= − (y; t q) L v L1 | because the series on both sides have a zero constant term. (k) ( 1) The case k > 1. Assuming that inv (t) = k− (y; t q) has already been proved • for k 1, the goal is to show L L | ≥ (k+1) ( 1) in (t)= − (y; t q). L v Lk+1 | Equating the coefficients of xk+1 in equation (2.14) yields ( 1) ( 1) ( 1) − (y; t q)= − (y; t q) − (y; t q). DqLk+1 | DqL1 | ·Lk | If we can show that similarly in(k+1)(t)= in(1)(t) in(k)(t) (3.3) DqL v DqL v ·L v then we would be done. Again the final integration step poses no problem because (k+1) ( 1) in both inv (t) and − (y; t q) the first k + 1 coefficients vanish. Recall that L Lk+1 | a configuration λ in(k+1)[n] consists of a list of k +1 disjoint strict lists, written as a list λ =(λ ,λ∈L,...,λ ), with λ in(1)[A ], where 0 1 k i ∈L i k
Ai = [n] and min Ai 1 < min Ai (1 i k). − ≤ ≤ i=0 ] We have a bijection (k+1) (1) (k) in [A] in [A ] in [A′]: λ (λ ,λ′) L →L 0 ×L 7→ 0 k where λ′ =(λ1, ,λk) and A′ = i=1Ai, which also satisfies the requirement for applying the folklore··· statement (3.1):∪ ′ inv(A0,A ) v(λ)= v(λ ) v(λ′) q . 0 · · All this holds only if for the bipartition A = A A′ it is guaranteed that min A < 0 ⊎ 0 min A′. This is where the derivative q comes into play. Derivation for collection of structures means that the minimumD element of the underlying set of a structure is tagged and no longer counted in the w-valuation of the base set. In the present situation that only structures are considered where tagging the minimum element S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 11
of λ means the same as tagging the minimum element of λ0. This shows that (3.3) holds. We hence complete the proof. Theorem 5. For integer α 1 we have ≥ − (α) σ +rl(λ) inv(σ.λ) rl(λ)+inv(σ) ℓn,k(y; q)= y| | q − . α (σ;λ) ( ) X∈LCn,k Proof. From (2.10) we derive
(α) (α) ( 1) (y; t q)= (0; y; t q) − (y; t q), Lk | L | Lk | and the result follows from Lemmas 1 and 2. Here we give an example to illustrate the α-Laguerre configurations. Example 1. Consider the 1-Laguerre configuration (σ; λ) (1) in Figure 2. We have ∈ LC15,4 σ = σ σ with σ = (15)(74), σ = (14)(1352); 0 · 1 0 1 λ =(λ1,λ2,λ3,λ4) with λ1 =13, λ2 = 12611, λ3 = 108, λ4 =9. Thus σ =σ ˆ σˆ = 7415 132514, 0 · 1 · σ =03 104; λ =13 12611 108 9. · · · We have σ =7, rl(λ)=3, inv(σ)=4, and inv(σ λ)=52. | | · Remark 3. Our model of α-Laguerre configurations is simpler than the model in [3]. Ac- tually, the α-Laguerre configurations are essentially the Laguerre configurations of which each cycle has a color in 0,...,α . A linear order of paths and colored cycles is needed only for the valuation w in{ (3.2). }
4. Moments of (q,y)-Laguerre polynomials For α N , by (2.6) the moments of the (q,y)-Laguerre polynomials are defined by ∈ 0 µ(α)(q,y) := (xn). (4.1) n Lq According to the theory of orthogonal polynomials (see [4]) and the three-term recur- rence relation (1.7), we have the orthogonality relation n (L(α)(x; y; q)L(α)(x; y; q)) = ynn! [α + j] δ . (4.2) Lq n m q q n m j=1 Y 12 QIONGQIONGPANANDJIANGZENG
Moreover, we have the following continued fraction expansion: 1 µ(α)(q,y)tn = , (4.3) n λ t2 n 0 1 b t 1 X≥ − 0 − λ t2 1 b t 2 1 . − − ..
where bn = y[n + α + 1]q + [n]q and λn = y[n]q[n + α]q. Let Sn be the set of permutations of 1, 2, , n . For σ Sn, we define three statistics, respectively, { ··· } ∈ the number of weak excedances, wex(σ), by • wex(σ)= i [n]: σ(i) i ; |{ ∈ ≥ }| the number of records (or left-to-right maxima), rec(σ), by • rec(σ)= i [n]: σ(i) > σ(j) for all j < i ; |{ ∈ }| the number of crossings, cros(σ), by • cros(σ)= (i, j) [n] [n]: i < j σ(i) < σ(j) or σ(j) < σ(i) (α) n 1 µn (q,y)t = , (4.5) γ1t n 0 1 X≥ γ2t − 1 . − .. n where γ2n = [n]q and γ2n+1 = y[n + α]q = y([n]q + [α + 1]qq ) for n 0. Recall that a Dyck path of length 2n is a sequence of points (ω≥,...,ω ) in N N 0 2n × satisfying ω0 = (0, 0), ω2n = (2n, 0) and ωi+1 ωi = (1, 1) or (1, 1) for i =0,..., 2n 1. Clearly we can also identify a Dyck path with− its sequence of− steps (or Dyck word− ) s = s ...s on the alphabet u, d and we use s u and s d to denote the number of u and 1 2n { } | | | | d respectively in s so for a Dyck word s, we have s u = s d = n and s1 ...sk u s1 ...sk d for k [2n]. The height h of step s is defined| to| be|h| = 0 and | | ≥| | ∈ k k 1 hk = s1 ...sk 1 u s1 ...sk 1 d for k =2,..., 2n. | − | −| − | S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 13 A Laguerre history of length 2n is a pair (s,ξ), where s is a Dyck word of length 2n and ξ =(ξ ,...,ξ ) is a sequence such that ξ = 1 if s = u and 1 ξ h /2 if s = d. Let 1 2n i i ≤ i ≤ ⌈ i ⌉ i n be the set of Laguerre histories of length 2n. We essentially use Biane’s bijection [2] LHto construct a bijection Φ from S to . n LHn Proof of Theorem 6. We identify a permutation σ S with the bipartite graph on ∈ n G 1,...,n;1′,...,n′ with an edge (i, j′) if and only if σ(i) = j. We display the vertices {on two rows called} top row and bottom row as follows: 1 2 n ··· 1′ 2′ n′ ··· and read the graph column by column from left to right and from top to bottom. In other words, the order of vertices is v1 =1, v2 =1′,...,v2n 1 = n, v2n = n′. − For k =1,..., 2n the k-th restriction of is the graph k on v1, v2,...,vk with edge (v , v ) in if and only if i, j [k], so isolatedG vertices mayG exist{ in . } i j Gk ∈ Gk For i =1,...,n the Dyck path s = s1 ...s2n is defined by the following: 1 if σ− (i) >i<σ(i) (i.e., i is a cycle valley), then s2i 1s2i = uu; • 1 − if σ− (i) Next, the sequence ξ =(ξ1,...,ξ2n) is defined as follows: si = u then ξi = 1; • s = d, then • i – if σ(i) < i (i.e., i is a cycle double descent or cycle peak), then h2i 1 > 0, − let ξi = m if σ(2i) is the m-th isolated vertex at the bottom row of 2i 2 G − from right-to-left (1 m h2i/2 ), clearly the value i will contribute m 1 crossings l Let Φ(σ)=(s,ξ). Then wex(σ)= i [n]: s = d , |{ ∈ 2i }| rec(σ)= i [n]: s = d, ξ = h /2 , |{ ∈ 2i 2i ⌈ 2i ⌉}| cros(σ)= (ξ 1), i − d i:Xsi= therefore, rec(σ) wex(σ) cros(σ) ξi 1 χ(ξ i= h i/2 ) β y q = q − yβ 2 ⌈ 2 ⌉ σ Sn (s,ξ) n i:si=d i:s2i=d X∈ X∈LH Y Y = w(si), (4.6) s Dyck i:si=d ∈X n Y where Dyckn denotes the set of Dyck paths of semilength n and the weight of each down step si = d is defined by k 1 1+ q + + q − if hi =2k, w(si)= ··· k 1 k y(1 + q + + q − + βq ) if h =2k +1. ( ··· i A folklore theorem [5] implies that the generating function of (4.6) has the continued fraction expansion (4.5) and we are done. Example 2. If σ =412796583 S9, then the Laguerre history Φ(σ)=(s,ξ) is given by ∈ s uu du du ud uu ud dd ud dd = . ξ 11 11 11 12 11 11 12 11 11 Theorem 7. Let α N . For nonnegative integers n ,...,n the linearization coefficient ∈ 0 1 k m (α) q Lnk (x; y; q) (4.7) L ! Yk=1 is a polynomial in N[y, q]. Proof. In view of the orthogonality (4.2) it suffices to prove the m = 3 case. Indeed, we can derive the following explicit formula from [17, Theorem 1]: (L(α)(x; y; q)L(α)(x; y; q)L(α)(x; y; q)) Lq n1 n2 n3 s s = n1!q n2!q n3!q y n1 + n2 + n3 2s,s n3,s n2,s n1 q s max(n1,n2,n3) (4.8) ≥ X − − − − − − α + s n1 + n2 + n3 2s k k+1 + n1+n2+n3 2s k +kα − y q( 2 ) ( 2 ) , × s k q k 0 q X≥ S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 15 where the q-multinomial coefficients a + b + c + d (a + b + c + d)! = q , a,b,c,d a! b! c! d! q q q q q are known to be polynomials in N[q] for integral a,b,c,d 0, see [13]. Hence, the right- hand side of (4.8) is a polynomial in N[y, q] and we are done.≥ For arbitrary α a combinatorial interpretation of (4.7) was given by Foata–Zeilberger [8] with y = q = 1, and generalized by the second author [23] to q = 1; see also [24], while for α = 0 a combinatorial interpretation of (4.7) was given by Kasraoui et al. [17]. Thus, the following problem is deemed interesting. Problem. What is the combinatorial interpretation of (4.7) for α N0 unifying the two special cases with α =0 or q =1? ∈ 5. Connection to rook polynomials and matching polynomials In this section we show the connection of the model of α-Laguerre configurations with the models of non-attacking rook placements and matchings of complete bipartite graphs. 5.1. Interpretation in rook polynomials. An m-by-n board B is a subset of an m n grid of cells (or squares). A rook is a chessboard piece which takes on rows and columns.× If rk is the number of ways of putting k non-attacking rooks on this board, then the ordinary rook polynomial is defined by k Rm,n(x)= rkx . Xk Thus the Laguerre polynomials (1.2) can be written as (α) n 1 L (x)=( 1) n!R ( x− ). (5.1) n − n,n+α − An k non-attacking rook placement on a board B is a subset C B of k cells such that no two cells are in the same row or column of B. We refer the reader⊂ to Riordan’s classical book [20, Chapters 7 and 8] for many problems formulated in terms of configurations of non-attacking rooks on ”chessboards” of various shapes. We label the grid in the row from top to bottom and the columns from left to right in the same way as referring to the entries of an m n matrix. Recall that an integer partition × is a sequence of positive integers µ := (µ1,µ2,...,µl) such that µ1 µ2 µl > 0. m1 mk ≥ ≥···≥ We also use the notation µ =(n1 ,...,nk ) to denote the partition with mi parts equal to ni for i =1,...,k. For convenience, we shall identify µ with its Ferrers bord Bµ, which is defined as the subset (i, j):1 i µ , 1 j l of N N. For a placement C { ≤ ≤ j ≤ ≤ } × of rooks on Bµ, the inversion number inv(C) is defined as follows: for each rook (cell) in C cross out all the cells which are below or to the right of the rook, then inv(C) is the number of squares of Fµ not crossed out. An example is depicted in Figure 3. 16 QIONGQIONGPANANDJIANGZENG × ×× Figure 3. The Ferrers board of shape µ = (4, 4, 3, 3, 1) and a placement C of three non-attacking rooks with inv(C) = 3. Definition 8. For integers n, k 0 and α 1, let m = (m0,...,mα) and n = (n ,...,n ) be non-negative integer≥ sequences such≥ − that m +m + +m +n + +n = 1 k 0 1 ··· α 1 ··· k n with m 0 and n 1. We define (α)(m; n) as the set of n n squares of color i ≥ j ≥ Bn,k × shape B := (B(1); B(2)) with B(1) :=(nm0 ,...,nmα ), (5.2a) B(2) :=(nn1 ,...,nnk ). (5.2b) By convention, if α = 1 (resp. k =0), then B(1) = (resp. B(2) = ). Let − ∅ ∅ α α cw(B)= m and cd(B)= i m . i · i i=0 i=0 X X (α) (α) Let n,k (m; n) denote the set of all ordered pairs = (B,C), where B n,k(m; n) and BCC is an n non-attacking rook placement on B suchR that ∈ B min(C B(2)) < min(C B(2)) < < min(C B(2)), (5.3) ∩ 1 ∩ 2 ··· ∩ k (2) (2) where min(C B1 ) is the minimum row index of cells in C B1 . For each block (2) ni ∩ (2) ∩ (2) Bi = (n ) we define ind(C Bi ) as the number of rooks in C Bi whose column indexes are greater than the column∩ index of the rook which has the∩ maximum row index in B(2) and let ind( )= k ind(C B(2)). Let i R i=1 ∩ i P α k (α) = (α)(m; n) with m + n = n. BCn,k BCn,k i j m,n i=0 j=1 [ X X An element =(B,C) (α) is called a colored rook configuration. R ∈ BCn,h Remark 4. One can imagine that each column of a board in (α)(m; n) is colored with BCn,k colors in 0, 1,...,α + k from left to right as follows: the first m columns get color 0, { } 0 the next m1 columns get color 1, ..., the last nk columns get color α + k. S´eminaire Lotharingien de Combinatoire 81 (2018), Article B?? 17 1 2 × 3 × 4 × 5 × 6 × 7 × 8× 9 × 10 × 11 × 12 × 13 × 14 × 15 × 1 2× 3 4 5 6 7 8 9 101112131415 Figure 4. The colored rook configuration corresponding to the 1- Laguerre configuration in Figure 2 with (m; n)R = ((3, 4);(2, 3, 2, 1)). (α) Theorem 9. The coefficient ℓn,k(y; q) in (1.8) is the following generating polynomial of colored rook configurations in (α) BCn,k (α) cw(B)+ind( ) inv(C)+cd(B) ind( ) ℓn,k(y; q)= y R q − R . α =(B,C) ( ) R X∈BCn,k Proof. Let (α)(m; n) be the set of ρ := (σ , , σ ; λ , ,λ ) (α) such that LCn,k 0 ··· α 1 ··· k ∈ LCn,k ρ = ( σ ,..., σ ; λ ,..., λ )=(m; n). We define the map φ : (α)(m; n) | | | 0| | α| | 1| | k| LCn,k −→ (α)(m; n) by φ(ρ)=(B,C) for ρ =(σ , , σ ; λ , ,λ ) (α)(m; n) as follows: BCn,k 0 ··· α 1 ··· k ∈ LCn,k (i) The colored board B =(B(1), B(2)) is given by (1) σ0 σα (2) λ1 λ2 λk B =(n| |,...,n| |) and B =(n| |, n| |,...,n| |). (ii) If w :=σ ˆ σˆ σˆ λ λ λ = w ...w , which is a permutation of [n], let C = 0 1 ··· α 1 2 ··· k 1 n (j, wj): j [n] . { ∈ } (α) It is clear that φ(ρ) n,k and the procedure is reversible. Hence φ is a bijection. It ∈ BC (2) α is easy to verify that inv(C) = inv(ρ), ind(Bi ) = rl(λi), and cd(B) = i=0 i σi , which implies that | | P σ + rl(λ)= cw(B)+ ind( ); | | R inv(σ.λ) rl(λ)+ inv(σ)= inv(C)+ cd(B) ind( ). − − R 18 QIONGQIONGPANANDJIANGZENG The result then follows from Theorem 5. Example 3. Let ρ = ((74)(15), (1325)(14);13, 12611, 108, 9) (1) , then φ maps it ∈ LC15,4 to the placement of 15 rooks on the board B = (157;152, 103, 82, 7), see Figure 4. We find cw(B)=1, cd(B) = 4; inv(C)=52 and ind( )=3. R 5.2. Interpretation in matching polynomials. Recall that a matching of a graph G is a set of edges without common vertices. For any graph G with n vertices, the matching polynomial of G is defined by n/2 ⌊ ⌋ k n 2k α(G, x)= ( 1) m x − . − k Xk=0 where mk is the number of k-edge matchings of G. Let Kn,m denote the set of complete bipartite graphs on the two disjoint sets A = [n] and B = 1′,...,m′ , that is, there is an edge (a, b) if and only if a A and b B. From the explicit{ formula} (1.2) it is quite easy to derive the connection formula∈ ∈ α(K , x)= xαL(α)(x2) (α 1). (5.4) n,n+α n ≥ − Godsil and Gutman [12] proved (5.4) by showing that the matching polynomials satisfy the same three-term recurrence relation (1.3). Here we give a simple bijection between our α-Laguerre configuration model and the above matching model of complete bipartite n k graph. Let − be the set of matchings of K with n k edges. Mn,m n,m − Proposition 10. For integers n, k 1 and α 1 there exists an explicit bijection (α) n k ≥ ≥ − φ : − . LCn,k −→ Mn,n+α Proof. We construct such a bijection φ. Let ρ =(σ , σ , , σ ; λ ,λ , ,λ ) (α) be 0 1 ··· α 1 2 ··· k ∈ LCn,k an α-Laguerre configuration. We define a matching γ of Kn,n+α such that (a, b′) A B is an edge in γ if and only if (a, b) satisfies one of the following three conditions:∈ × (1) σ0(a)= b, i.e., the image of a is b through the action of permutation σ0; (2) a and b are consecutive letters in the wordσ ˆ1(n + 1)ˆσ2(n + 2) ... σˆα(n + α); (3) a and b are consecutive letters in the word λ for some j [k]. j ∈ By convention, if α = 1 (resp. 0) there are no words of types (1) and (2) (resp. type (2)). Since σ σ ...σ λ−...λ is a permutation of [n] it is clear that there are n k such 0 1 α 1 k − edges (a, b′). The above procedure is obviously reversible. Example 4. For the 1-Laguerre configuration ρ = ((74)(15), (1325)(14);13, 12 6 11, 10 8, 9) in (1) from Example 1, the corresponding matching γ of K11 is depicted in Figure 2. LC15,4 15,16 Remark 5. We leave the interested reader to find the (q,y)-version of the above matching (α) polynomials for (q,y)-Laguerre polynomials Ln (x; y; q). 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Zeng, Combinatorics of orthogonal polynomials and their moments, Lecture Notes at OPSF Summer School, Univ. Maryland, College Park, MD, July 11–July 15, 2016. Univ Lyon, Universite´ Lyon1, UMR 5208 du CNRS, Institut Camille Jordan, F-69622, Villeurbanne Cedex, France E-mail address: [email protected] Univ Lyon, Universite´ Lyon1, UMR 5208 du CNRS, Institut Camille Jordan, F-69622, Villeurbanne Cedex, France E-mail address: [email protected]