(q,t)-Catalan numbers: gamma expansions, pattern avoidances, and the (−1)-phenomenon

Fu, Shishuo; Tang, Dazhao; Han, Bin; Zeng, Jiang

doi:10.1016/j.aam.2019.01.009

Cited by 8 publications

(5 citation statements)

References 38 publications

(105 reference statements)

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“…The following relation parallels (2.2), and has previously been observed in [13,Lemma 5.4]. For any permutation p P S n , 231ppq ´213ppq " n ´pn ´desppq " MAJppq ´BASTppq.…”

Section: Moreover Letsupporting

confidence: 79%

From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

Chen

2019

Advances in Applied Mathematics

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“…The following relation parallels (2.2), and has previously been observed in [13,Lemma 5.4]. For any permutation p P S n , 231ppq ´213ppq " n ´pn ´desppq " MAJppq ´BASTppq.…”

Section: Moreover Letsupporting

confidence: 79%

From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

Chen

2019

Advances in Applied Mathematics

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“…Fu et al [14] gave more interpretations of N n (t/q, q, 1) and N n (t, q, t) in terms of npermutation patterns. We further prove the following interpretations by using the (n − 1)permutation patterns.…”

Section: Resultsmentioning

confidence: 99%

“…(3.41) Remark 3.11. Other interpretations for γ n−1,k (q) and γ n−1,k−1 (q) are given in [14,16,17].…”

Section: Resultsmentioning

confidence: 99%

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Eulerian polynomials and excedance statistics

Han

Mao

Zeng

2019

Preprint

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A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. Rephrasing the latter formula with permutation cycle peaks and excedances we are able to prove a series of general formulas expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulas are comparable with Zhuang's generalizations [Adv. in Appl. Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng [European J. Combin. 33 (2012), no. 2, 111-127] and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the γ-coefficients of the inversion polynomials on 321-avoiding permutations.

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“…Since π rc P S n,k p132q if and only if π P A n,k p213q, then #A n,k p213q " #A n,k p132q. Notice that π is 213-avoiding if and only if π is 213-avoiding, see Lemma 2.8 in [10]. Hence, #A n,k p213q " #A n,k p213q.…”

Section: A Nk P213q and Distribution Of Des Over A Nk P213qmentioning

confidence: 96%

Statistics on permutations with bounded drop size

Chen,

Zhou

2021

Preprint

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Permutations with bounded drop size, which we also call bounded permutations, was introduced by Chung, Claesson, Dukes and Graham. Petersen introduced a new Mahonian statistic the sorting index, which is denoted by sor. Meanwhile, Wilson introduced the statistic DIS, which turns out to satisfy that sorpσq " DISpσ ´1q for any permutation σ. In this paper, we maintain Petersen's method to deduce the generating functions of pinv, lmaxq and pDIS, cycq over bounded permutations to show their equidistribution. Moreover, the generating function of des over 213-avoiding bounded permutations and some related equidistributions are given as well.

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(q,t)-Catalan numbers: gamma expansions, pattern avoidances, and the (−1)-phenomenon

Cited by 8 publications

References 38 publications

From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

From q-Stirling numbers to the ordered multiset partitions: A viewpoint from vincular patterns

Eulerian polynomials and excedance statistics

Statistics on permutations with bounded drop size

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