Bin Han

University of Gothenburg | GU · Department of Oncology

We introduce a kind of $(p, q, t)$-Catalan numbers of Type A by generalizing the Jacobian type continued fraction formula, we proved that the corresponding expansions could be expressed by the polynomials counting permutations on $\S_n(321)$ by various descent statistics. Moreover, we introduce a kind of $(p, q, t)$-Catalan numbers of Type B by gen...

The binomial Eulerian polynomials, first introduced in work of Postnikov, Reiner and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations and proved that they can be written as the sums of two γ-positive...

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more equidistributions of mesh pattern and confirm Kitaev and Zhang's four conjectures by constructing two involutions...

The γ-coefficients of Eulerian polynomials were first considered by Foata and Schützenberger. In this paper, we provide combinatorial interpretations for the γ-coefficients arising from the segmented permutations and segmented derangements via Brändén’s modified Foata–Strehl action. We also give the combinatorial interpretations of γ-coefficients f...

In a recent paper ({arXiv:2101.01928v1}) Baril and Kirgizov posed two conjectures on the equidistibution of $(cyc, des_2)\sim (cyc, pex)$ and $(des, des_2)\sim (exc, pex)$, where cyc, des and exc are classical statistics counting the numbers of cycles, descents and excedances of permutation, while $des_2$ and $pex$ are numbers of special descents a...

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting p...

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more equidistributions of mesh pattern and confirm Kitaev and Zhang's four conjectures by constructing two involutions...

It is well known that the permutation peak polynomials and descent polynomials are connected via a quadratic transformation. By 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 E...

Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction exp...

The gamma positivity of a combinatorial sequence unifies both unimodality and symmetry. Finding new family of objets whose enumerative sequences have gamma positivity is a challenge and important topic in recent years. it has received considerable attention in recent times because of Gal’s conjecture, which asserts that the gamma-vector has nonnega...

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...

The aim of this paper is two-fold. We first prove several new interpretations of a kind of (q,t)-Catalan numbers along with their corresponding γ-expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain (−1)-phenomenon for each subset of permutations avoiding a single pattern of length three, and disc...

Motivated by the λ-Euler’s difference table of Eriksen et al. and colored Euler’s difference table of Faliharimalala and Zeng, we study the λ-analogue of colored Euler’s difference table and generalize their results. We generalize the number of permutations with k-excedances studied by Liese and Remmel in colored permutations. We also extend Wang e...

The aim of this paper is two-folded. We first prove several new interpretations of a kind of $q$-Narayana polynomials along with their corresponding $\gamma$-expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain $(-1)$-phenomenon for all Catalan subsets avoiding a single pattern of length three, an...