Eulerian quasisymmetric functions

Shareshian, John; Wachs, Michelle L.

doi:10.1016/j.aim.2010.05.009

Cited by 63 publications

(186 citation statements)

References 60 publications

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“…This result along with q-analogs were obtained independently by several authors; see e.g. [4,16,19,21]. Roselle [18] obtained an analogous result of Eq.…”

Section: Introductionsupporting

confidence: 64%

Signed countings of types B and D permutations and t,q-Euler numbers

Hsu

et al. 2018

Advances in Applied Mathematics

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It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length n is the Euler number En, alternating in sign, if n is odd (even, respectively). Josuat-Vergès obtained a q-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès and Kim.Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Vergès derived bivariate polynomials Qn(t, q) and Rn(t, q) as generalized Euler numbers via successive q-derivatives and multiplications by t on polynomials in t. The other goal in this paper is to give a combinatorial interpretation of Qn(t, q) and Rn(t, q) as the enumerators of the snakes with restrictions.

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“…This result along with q-analogs were obtained independently by several authors; see e.g. [4,16,19,21]. Roselle [18] obtained an analogous result of Eq.…”

Section: Introductionsupporting

confidence: 64%

Signed countings of types B and D permutations and t,q-Euler numbers

Hsu

et al. 2018

Advances in Applied Mathematics

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“…The incomparability graph of P is the path G n,2 " 1´2´¨¨¨´n. Stanley [7] obtained the formula for the generating function and Haiman [5] showed that X G n,2 px, tq is e-positive and e-unimodal. (2) Let m " pm 1 , m 2 , n, .…”

Section: Definition 22 ([8]mentioning

confidence: 99%

On $e$-Positivity and $e$-Unimodality of Chromatic Quasi-symmetric Functions

Cho¹

2019

SIAM J. Discrete Math.

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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. The e-positivity of chromatic quasisymmetric functions implies the e-positivity of corresponding chromatic symmetric functions, and our work resolves Stanley's conjecture on chromatic symmetric functions of p3`1q-free posets for two classes of natural unit interval orders.2010 Mathematics Subject Classification. Primary 05E05; Secondary 05C15, 05C25 .

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“…Next we give the definition of the statistic admissible inversion, which was first introduced by Shareshian and Wachs [37]. Definition 3.2.…”

Section: 2mentioning

confidence: 99%