With Advisor: Professor Jiang Zeng. Published in Université Claude Bernard Lyon 1, 2026
With Advisor: Professor Zhicong Lin. Published in Shandong University, 2022
With Jiang Zeng. Published in Advances in Applied Mathematics, 2025
Motivated by recent work on (re)mixed Eulerian numbers, we study a subfamily of these numbers introduced by Nadeau and Tewari. We establish a combinatorial interpretation by showing that they arise as generating polynomials of permutations with respect to a multivariate statistic involving left-to-right minima, right-to-left minima, descents, and the mixed major index. Our results simultaneously generalize the bi-Stirling-Eulerian polynomials of Carlitz–Scoville and the Stirling-Euler-Mahonian polynomials of Butler.
With Qiongqiong Pan and Jiang Zeng. Published in Enumerative Combinatorics and Applications, 2024
Motivated by recent works on enumeration of Coxeter groups by the parity of descent positions, we prove a formula for the generating function of the vector statistic \((\operatorname{odes}_G,\operatorname{edes}_G,\operatorname{col}_G,\ell_G)\) over the group of colored permutations \(G(r,n)\). This generalizes and unifies several known results over Coxeter groups of type \(A\) and \(B\).
With Zhicong Lin and Tongyuan Zhao. Published in European Journal of Combinatorics, 2022
We study the descent polynomials \(A_n^{(p)}(t)\) and \(B_n^{(p)}(t)\) associated with permutations of the multisets \(\{1^p,2^p,\ldots,n^p\}\) and \(\{1^p,2^p,\ldots,n^p,n+1\}\). We give a combinatorial interpretation for the \(\gamma\)-coefficients of \(A_n^{(p)}(t)\) via weakly increasing trees, answering a recent open problem of Lin–Ma–Ma–Zhou. We also prove that \(B_n^{(p)}(t)\) admits a bi-\(\gamma\)-positive expansion \(B_n^{(p)}(t)=a_n(t)+t b_n(t)\), where \(b_n(t)=(p-1)A_n^{(p)}(t)\), extending a result of Ma–Ma–Yeh from \(p=2\) to arbitrary \(p\).
With Jiang Zeng. Published in 37th Conference on Formal Power Series and Algebraic Combinatorics (Sapporo), 2025
We define a multivariable generalization of the Eulerian polynomials using linear and descent based statistics of permutations and establish the connection with the \((\alpha,t)\)-Eulerian polynomials based on cyclic and exceedance based statistics of permutations. As applications of this connection, we obtain the exponential generating function for the multivariable Eulerian polynomials and \(\gamma\)-positive formulas of two variants of Eulerian polynomials. We also show that enumerating the cycle André permutations with respect to the number of drops, fixed points and cycles gives rise to the normalised \(\gamma\)-vectors of the \((\alpha,t)\)-Eulerian polynomials. Our result generalizes and unifies several recent results in the literature.
With Sergey Kitaev and Dun Qiu. Submitted to ...
A boxed mesh pattern is a mesh pattern whose selected entries lie in an empty axis-parallel rectangle. We classify coincidences of boxed patterns with classical and vincular patterns, exhibit a genuinely bivincular coincidence, and prove that no boxed–bivincular coincidence occurs for patterns of length at least five. Together with known results, this shows that every boxed pattern of length at least five has factorial growth and hence fails the Stanley–Wilf property. At length four, one exceptional orbit is enumerated by the semi-Baxter numbers, while the remaining exceptional orbit, {2143,3412}, is unresolved; we conjecture that it has factorial growth. For Box(123), we derive an exact maximum-insertion identity and prove the subfactorial upper bound \(2^{5n}n^{\beta n}\), where \(\beta=\log_2(2\cos(\pi/7))<0.85\). A closed enumeration remains open. We also prove a general first-moment formula for boxed mesh patterns that depends only on the length of the underlying pattern; in particular, the expected number of Box(123) occurrences in a uniformly random permutation of length \(n\) is asymptotic to \(n\log n\).
With Yao Dong. Submitted to Discrete Applied Mathematics
Motivated by a recent conjecture of Bsila, Cox, Hugo, Styron and Zhuang concerning fixed points and pixed points on pattern-avoiding permutations, we prove a bivariate refinement involving descent statistics. Given a set of permutations \(\Pi\), let \(S_n(\Pi)\) denote the set of permutations in the symmetric group \(S_n\) that avoid every element of \(\Pi\) in the sense of pattern avoidance. For each set \(\Pi\) appearing in their conjecture, we show that the pairs of statistics \((\operatorname{des},\operatorname{fix})\) and \((\operatorname{ides},\operatorname{pix})\) are equidistributed over \(S_n(\Pi)\). Our proof is based on explicit ordinary generating functions for the corresponding pattern-avoiding classes.
With Jiang Zeng. Accepted by Combinatorial Theory, 2026
In 1977, Carlitz and Scoville introduced the cycle \((\alpha,t)\)-Eulerian polynomials by enumerating permutations with respect to the number of excedances, drops, fixed points, and cycles. In this paper, we propose a nine-variable generalization of the Eulerian polynomials defined in terms of descent-based statistics of permutations, and we establish a connection formula linking these two families of generalized Eulerian polynomials. By means of this connection formula, we derive explicit exponential generating functions and several \(\gamma\)-positive expressions for various Eulerian-type polynomials. Our results unify and strengthen recent findings of Ji and Ji–Lin. We also show that enumerating web permutations with respect to the numbers of drops, fixed points, and cycles yields the normalized \(\gamma\)-vectors of the \((\alpha,t)\)-Eulerian polynomials.
With Jiang Zeng. Submitted to Discrete Mathematics
We establish Jacobi-type continued fraction expansions for \((p,q)\)-analogues of the bi-Stirling Eulerian polynomials, which refine classical Eulerian statistics on permutations. These polynomials admit equivalent definitions in terms of descents or excedances. The parameters \(p\) and \(q\) encode refined permutation statistics, including \((\mathrm{des},\mathrm{asc})\) and \((\mathrm{cros},\mathrm{nest})\). Our approach is based on weighted Motzkin path models arising from variants of the bijections of Françon–Viennot and Foata–Zeilberger. As applications, we obtain new results on total positivity and \(\gamma\)-positivity for several families of enumerative polynomials. Our results unify and extend a number of recent works in the literature.
Submitted to Annals of Combinatorics
In this paper, we introduce two families of partial Laguerre histories by imposing natural restrictions on the labels of matched north-east and south-east steps. We show that these two families share the same Jacobi continued-fraction expansion. As applications, we derive new combinatorial interpretations of the Euler numbers and the reduced tangent numbers by excluding certain colored east steps. We also introduce a natural \((p,q)\)-weight on partial Laguerre histories and obtain \((p,q)\)-analogues of Euler numbers and reduced tangent numbers.
Submitted to Discrete Mathematics
Using decorated permutations and Ji’s sign-reversing involution, we give a combinatorial interpretation of Désarménien’s formula for the distribution of the reduced major index over derangements. In particular, we show that the main families contribute uniformly to all residue classes modulo \(n\), while a unique exceptional object produces the correction term.