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P-Partitions and P-Positivity Séminaire Lotharingien de Combinatoire 82B (2019) Proceedings of the 31st Conference on Formal Power Article #61, 12 pp. Series and Algebraic Combinatorics (Ljubljana) P-partitions and p-positivity Per Alexandersson∗ and Robin Sulzgrubery Dept. of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden Abstract. Using the combinatorics of a-unimodal sets, we establish two new re- sults in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse P-partitions expand positively into quasisymmetric power sums. Consequently any nonnegative linear combination of such functions is p-positive whenever it is symmetric. We apply this method to derive positivity results for chromatic quasisymmetric functions and unicellular LLT polynomials. Keywords: posets, symmetric functions, unimodality 1 Introduction Whenever a new family of symmetric functions is discovered, one of the most logical first steps to take is to expand them in one of the many interesting bases of the space of symmetric functions. This paradigm can be traced from Newton’s identities to modern textbooks such as [17]. Of special interest are expansions in which all coefficients are nonnegative integers. Such coefficients frequently encode highly nontrivial combinato- rial or algebraic information. A symmetric function is called p-positive if the expansion in the power-sum basis has nonnegative coefficients. There are numerous results in the literature regarding p- positivity, see for example [20, 19,4, 21,8,2]. In particular, the expansion of a symmetric function into power sum symmetric functions can be useful when one is working with plethystic substitution [15], or evaluating certain polynomials at roots of unity [7, 19]. In this extended abstract we provide a uniform method for finding power sum ex- pansions and proving p-positivity in many of the cases mentioned above. Our approach requires a detour to quasisymmetric functions. There is a quasisymmetric extension of p-positivity, namely positivity in a quasisymmetric power-sum basis. Quasisymmetric power sums were recently introduced by C. Ballantine et al. [5]. They appear as duals of noncommutative power sums defined by I. Gelfand et al. [9]. ∗[email protected]. Per Alexandersson was supported by Knut and Alice Wallenberg Foun- dation (2013.03.07). [email protected]. 2 Per Alexandersson and Robin Sulzgruber Our first main result, Theorem 2.2, is an expansion of the fundamental quasisymmet- ric functions Fn,S into quasisymmetric power sums Ya: ( ) Ya x jSnSaj Fn,S(x) = ∑ (−1) a za Here the sum ranges over all compositions a of n such that the set S is a-unimodal, and za and Sa denote an integer factor and a set associated to a. All definitions are given in Section2. Our second main result, Theorem 3.2, is the following statement: Let P be a finite poset on n elements and KP(x) be the generating function of order-preserving maps f : P ! N+, so called P-partitions. Then Ya ∗ Ya ∗ KP(x) = ∑ jLa(P, w)j = ∑ jOa(P)j . za za an an ∗ Here w denotes an arbitrary natural labeling of P, La(P, w) consists of certain a-unimodal ∗ linear extensions of P, and Oa(P) is a set of certain order-preserving surjections from P onto a chain. These definitions are found in Section3. As a consequence, we can produce an expansion into power sums for any symmet- ric function, for which the expansion in Gessel’s fundamental basis or in terms of P- partitions is known. Moreover any symmetric function which is a nonnegative linear combination of functions KP is automatically p-positive. As a bonus it now becomes apparent that p-positivity of certain families of symmetric functions is really a special case of a more general positivity phenomenon, namely Y-positivity, that encompasses a larger class of quasisymmetric functions. In Section 4 we give a few examples of the numerous implications of the above formula. The full paper [3] contains additional ap- plications and results. Some exciting recent result related to this extended abstract are due to by R. Liu and M. Weselcouch [14]. 2 The Y-expansion of fundamental quasisymmetric func- tions In this section we briefly introduce the space of quasisymmetric functions and a few relevant bases. For a more thorough background on quasisymmetric functions, we refer the reader to the references [23, 16]. Afterwards, we define a-unimodality and present the expansion of the fundamental basis into quasisymmetric power sums. P-partitions and p-positivity 3 2.1 Quasisymmetric functions The monomial quasisymmetric function Ma, where a is a composition with ` parts, is de- fined as a1 a2 a` Ma(x) := x x ··· x . ∑ i1 i2 i` i1<i2<···<i` The functions Ma constitute a basis for the space of homogeneous quasisymmetric func- tions of degree n as a ranges over all compositions of n. Given a composition a n with ` parts, let Sa := fa1, a1 + a2,..., a1 + a2 + ··· + a`−1g. The map a 7! Sa defines the usual bijection between compositions of n and subsets of [n − 1]. We let a ≤ b denote the fact that a is a refinement of b, that is, b can be obtained from a by adding consecutive parts of a. Whenever a ≤ b, we can illustrate this relationship with bars between parts of a, such that parts between bars add to parts of b. For example 112j341j21j34j2 corresponds to a = 11234121342, b = 48372. Finally, given a permutation s 2 Sn and compositions a ≤ b, we can partition s into a-subwords and b-blocks of words. For example, a = 3132, b = 45, s = 926345817 is partitioned as (926)(3)j(458)(17), (2.1) where the first b-block is (926)(3), consisting of two a-subwords. The fundamental quasisymmetric function Fn,S can be defined in two equivalent ways: ( ) := ··· ( ) := ( ) Fn,S x ∑ xj1 xjn or Fa x ∑ Mb x , j1≤j2≤···≤jn b≤a i2S)ji<ji+1 where Fn,Sa (x) and Fa(x) are equal for all compositions a n. Given a pair of compositions of n, a ≤ b, related by j j · · · j a11a12 ... a1,i1 a21a22 ... a2,i2 ak1ak2 ... ak,ik let k ( ) := ( )( + ) ··· ( + + ··· + ) p a, b ∏ aj1 aj1 aj2 aj1 aj2 aj,ij . j=1 4 Per Alexandersson and Robin Sulzgruber 1 The quasisymmetric power sum Ya is defined as 1 Y (x) := z M (x). a a ∑ ( ) b (2.2) b≥a p a, b 1 1 3 For example, Y231 = 10 M6 + 4 M24 + 5 M51 + M231. It was shown in [5, Thm. 3.11] that quasisymmetric power sums refine the usual power sums as pl(x) = ∑ Ya(x) a∼l where the sum is taken over all compositions a that are a permutation of l. Let w be the involution on quasisymmetric functions that sends Fn,S to F[n−1]n(n−S). This extends the classical involution on symmetric functions, for which whl = el, and jlj−`(l) jaj−`(a) w(pl) = (−1) pl. Then (see [5, Sec. 4]) we have that w (Ya) = (−1) Yar , where ar denotes the reverse of a. 2.2 The Y-expansion of the fundamental basis A word s1s2 ··· sn is said to be unimodal if there is a k 2 [n] such that s1 > ··· > sk < ··· < sn. Given a permutation s 2 Sn and a composition a n, we can partition s into nonover- lapping subwords with sizes given by a. A permutation s is a-unimodal if each subword determined by a is unimodal. Finally, a subset of [n − 1] is said to be a-unimodal if it is the descent set of some a-unimodal permutation in Sn. There are various equiva- lent characterizations of a-unimodal sets, and their properties are an interesting topic in itself. Example 2.1. Let a = 3513. Then the permutation s = 7, 2, 3, 12, 9, 8, 6, 11, 4, 1, 5, 10 is a- unimodal as the the four subwords 7, 2, 3 j 12, 9, 8, 6, 11 j 4 j 1, 5, 10 are unimodal. Hence the set DES(s) = f1, 4, 5, 6, 8, 9g is a-unimodal. The first main result is the following theorem: Theorem 2.2. Let n 2 N and S ⊆ [n − 1]. Then ( ) Ya x jSnSaj Fn,S(x) = ∑ (−1) , (2.3) a za where the sum ranges over all compositions a of n such that S is a-unimodal. 1 mi Here, za is the standard quantity ∏i≥1 i mi!, with mi being the number of parts of a equal to i. P-partitions and p-positivity 5 Sketch of proof. Expand both sides in the monomial basis and compare coefficients of Mb(x). It then suffices to prove that ( (−1)jSgnSaj 1 if b ≤ g, = (2.4) ∑ p(a, b) a2R(b,g) 0 otherwise, where R(b, g) is the set of all compositions a ≤ b such that Sg is a-unimodal. We now need some additional terminology from [5]. Let s be a permutation and a ≤ b be compositions. Consider the partitioning of s into a-subwords and b-blocks of subwords according to (2.1). We say s 2 Sn is consistent with a ≤ b if the following conditions are satisfied: (i) In each a-subword the maximum appears in last position. (ii) The subwords in each b-block are sorted increasingly with respect to their maxima. Let Cons(a, b) denote the set of permutations s 2 Sn that are consistent with a ≤ b. For example, the permutation (4)(38)j(7)j(56)(219) lies in Cons(12123, 3155), but the permutations (4)(38)j(7)j(65)(219) and (4)(38)j(7)j(59)(216) do not.
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