Séminaire Lotharingien de Combinatoire 80B (2018) Proceedings of the 30th Conference on Formal Power Article #59, 12 pp. Series and Algebraic Combinatorics (Hanover) On e-positivity and e-unimodality of chromatic quasisymmetric functions Soojin Cho∗1 and JiSun Huhy1 1Department of Mathematics, Ajou University, Suwon 16499 Republic of Korea Abstract. The e-positivity conjecture and the e-unimodality conjecture of chromatic quasisymmetric functions are proved for some classes of natural unit interval orders. Recently, J. Shareshian and M. Wachs introduced chromatic quasisymmetric functions as a refinement of Stanley’s chromatic symmetric functions and conjectured the e- positivity and the e-unimodality of these functions. Our work resolves the Stanley’s conjecture on chromatic symmetric functions of p3 ` 1q-free posets for two classes of natural unit interval orders. Résumé. La conjecture d’e-positivité ainsi que celle d’e-unimodalité sur les fonctions chromatiques quasi-symétriques ont été démontrées pour quelques classes d’ordres naturels sur l’ensemble des intervalles unitaires. Récemment, J. Shareshian et M. Wachs ont introduit les fonctions chromatiques quasi-symétriques comme un raffine- ment des fonctions chromatiques symétriques, et ont conjecturé qu’elles sont e-positives et e-unimodales. L’e-positivité d’une fonction chromatique quasi-symétrique implique celle de la fonction chromatique symétrique correspondante, et ce travail résout donc la conjecture de Stanley pour les fonctions chromatiques symétriques des ensembles ordonnés sans p3 ` 1q pour deux ordres naturels sur les intervalles unitaires. Keywords: chromatic quasisymmetric function, e-positivity, e-unimodality, p3 ` 1q-free poset, natural unit interval order 1 Introduction In 1995, R. Stanley [8] introduced the chromatic symmetric function XGpxq associated with any simple graph G, which generalizes the chromatic polynomial cGpnq of G. One of the long standing and well known conjecture due to Stanley on chromatic symmetric functions states that a chromatic symmetric function of any p3 ` 1q-free poset is a linear sum of elementary symmetric function basis telu with nonnegative coefficients. Recently, Shareshian and Wachs [7] introduced a chromatic quasisymmetric refinement XGpx, tq of chromatic symmetric function XGpxq for a graph G. They conjectured the e-positivity ∗[email protected]. This work is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01057476). [email protected] 2 Soojin Cho and JiSun Huh and the e-unimodality of the chromatic quasisymmetric functions of natural unit interval m j orders: That is, if XGpx, tq “ j“0 ajpxqt for the incomparability graph G of a natural unit interval order, then a pxq, 0 ¤ j ¤ m, is a nonnegative linear sum of e ’s, which is j ř l a refinement of the Stanley’s conjecture. Moreover, e-unimodality conjecture states that m´1 aj`1pxq ´ ajpxq is a sum of el’s with nonnegative coefficients for 0 ¤ j ă 2 . In this paper, we give combinatorial proofs of the conjectures on e-positivity and e- unimodality of chromatic quasisymmetric functions for certain classes of natural unit interval orders. We use the Schur function expansion of chromatic quasisymmetric func- tions in terms of Gasharov’s P-tableaux ([2]), that was done by Shareshian and Wachs in [7]. We use the Jacobi–Trudi expansion of Schur functions into elementary symmetric functions to write chromatic quasisymmetric functions as a sum of elementary symmet- ric functions with coefficients of signed sum of positive t polynomials. We then define a sign reversing involution to cancel out negative terms and obtain only positive terms. We use an inductive argument on P-tableaux to find explicit e-expansion formulae of XGpx, tq and this shows the e-unimodality. We remark that one class of natural unit interval orders we consider (in Section 3.1) was considered by Stanley–Stembridge and recently by Harada–Precup and the e-positivity of the corresponding chromatic symmet- ric functions was proved (Remark 4.4 in [11], [6]). 2 Preliminaries 2.1 Chromatic quasisymmetric functions and the positivity conjecture. We set P “ t1, 2, . u and rns “ t1, 2, . , nu. For n P P, a partition l “ pl1,..., l`q of n is a sequence of positive integers such that li ¥ li`1 for all i and i li “ n. For a 1 1 1 1 partition l, the conjugate of l is the partition l “ pl ,..., l q with l “ |ti | li ¥ ju|. 1 l1 řj For n P P, the nth elementary symmetric function e is defined as e “ x ¨ ¨ ¨ x , n n i1㨨¨ăin i1 in and the Q-algebra LQ of symmetric functions is the subalgebra of Qrrx1, x2,... ss generated 8 n n ř by the en’s; LQ “ L “ Qre1, e2,... s . Then L “ n“0 L , where L is the subspace of n symmetric functions of degree n, and tel | l $ nu is a basis of L , where el “ el1 ¨ ¨ ¨ el` . À n The set of Schur functions ts | l $ nu forms another basis of L , where s “ detre 1 s l l li´i`j is a determinant of a l1 ˆ l1 matrix, that is called Jacobi–Trudi identity.A proper coloring of a simple graph G “ pV, Eq is a function k : V Ñ P satisfying kpuq ‰ kpvq for any u, v P V such that tu, vu P E. For a proper coloring k, ascpkq “ tti, ju P E | i ă j and kpiq ă kpjqu. Definition 2.1 ([7]). For a simple graph G “ pV, Eq which has a vertex set V Ă P, the chro- matic quasisymmetric function of G is a sum over all proper colorings of G: ascpkq XGpx, tq “ t xk. k ¸ On e-positivity and e-unimodality of chromatic quasisymmetric functions 3 Note that chromatic quasisymmetric function XGpx, tq is a refinement of Stanley’s chromatic symmetric function XGpxq introduced in [8]; XGpx, tq|t“1 “ XGpxq. The incomparability graph incpPq of a poset P is a graph which has as vertices the ele- ments of P, with edges connecting pairs of incomparable elements. Natural unit interval orders are the posets we are interested in. Definition 2.2 ([7]). Let m :“ pm1, m2,..., mn´1q be a list of integers satisfying i ¤ mi ¤ mi`1 ¤ n for all i. The corresponding natural unit interval order Ppmq is the poset on rns with the order relation given by i ăPpmq j if i ă n and j P tmi ` 1, mi ` 2, . , nu. 1 2n Note that Catalan number Cn “ n`1 n counts the natural unit interval orders with n elements. Shareshian and Wachs showed that if G is the incomparability graph of a ` ˘ natural unit interval order then XGpx, tq is a polynomial with very nice properties. They also made a conjecture on the e-positivity and the e-unimodality of XGpx, tq. Remember n that a symmetric function f pxq P L is b-positive if the expansion of f pxq in the basis tblu n has nonnegative coefficients when tbl | l $ nu is a basis of L . Theorem 2.3 (Theorem 4.5 and Corollary 4.6 in [7]). If G is the incomparability graph of i a natural unit interval order then the coefficients of t in XGpx, tq are symmetric functions and |E| ´1 form a palindromic sequence in the sense that XGpx, tq “ t XGpx, t q. Conjecture 2.4 ([7]). If G is the incomparability graph of a natural unit interval order, then m i XGpx, tq is e-positive and e-unimodal. That is, if XGpx, tq “ i“0 aipxqt then aipxq is e-positive for all i, and a pxq ´ a pxq is e-positive whenever 0 ¤ i ă m´1 . i`1 i ř2 A finite poset P is called pr ` sq-free if P does not contain an induced subposet isomor- phic to the direct sum of an r element chain and an s element chain. Due to the result by Guay-Paquet [4], Conjecture 2.4 specializes to the famous e-positivity conjecture on the chromatic symmetric functions of Stanley and Stembridge: Conjecture 2.5 ([8, 11]). If a poset P is p3 ` 1q-free, then XincpPqpxq is e-positive. Since el is s-positive, Conjecture 2.4 implies the s-positivity of XGpx, tq, that was proved using the notion of P-tableaux by Shareshian–Wachs and Gasharov (t “ 1): Definition 2.6 (Gasharov [2]). Given a poset P with n elements and a partition l of n, a P-tableau of shape l is a filling T “ rai,js of a Young diagram of shape l in English notation a1,1 a1,2 ¨ ¨ ¨ a2,1 a2,2 ¨ ¨ ¨ . with all elements of P so that ai,j ăP ai,j`1 and ai`1,j ­ăP ai,j for all i and j. 4 Soojin Cho and JiSun Huh Definition 2.7 ([7]). For a finite poset P on a subset of P and a P-tableau T, let G be the incomparability graph of P. An edge ti, ju P EpGq is a G-inversion of T if i ă j and i appears below j in T. We let invGpTq be the number of G-inversions of T. The following theorem is the s-positivity result for the chromatic quasisymmetric functions of natural unit interval orders, which specializes to Gasharov’s s-positivity result for chromatic symmetric functions of p3 ` 1q-free posets. Theorem 2.8 ([7],[2]). Let G be the incomparability graph of a natural unit interval order P. invGpTq If we let lpTq be the shape of T, then XGpx, tq “ T t slpTq, where the sum is over all P-tableaux and therefore, X px, tq is s-positive.