Philip B. Zhang

Zeros and interlacing for multiset Eulerian-Narayana polynomials

Philip B. Zhang, Tongyuan Zhao

We prove the real-rootedness conjecture of Lin, Ma, Ma, and Zhou (2021) for multiset Eulerian-Narayana polynomials, including simplicity and negativity of all zeros after removing…

Infinite log-concavity of the Boros--Moll sequences

Matthew H. Y. Xie, Philip B. Zhang

Let $(d_{i} (n))_{i = 0}^{n}$ be the Boros--Moll coefficient sequence. We prove that, for every integer $n \geq 1$ , the polynomial \[ M_n(x)=\sum_{i=0}^n \bigl(d_i(n)^2-d_{i-1}(n)d_{i+1}(n)\…

Real-rootedness of the $Z$ -polynomials of sparse paving matroids and strict interlacing for uniform matroids

Matthew H. Y. Xie, Philip B. Zhang

We prove that the $Z$ -polynomial of every sparse paving matroid has only negative real zeros, confirming the real-rootedness conjecture of Proudfoot, Xu, and Young for this class.…

Real-rootedness of the $τ$ -polynomial under graph joins

Mingyang Kang, Zhixin Liu, Sophie C. C. Sun +1

For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $χ_{G} (x) = i = 0 \sum n (- 1)^{n - i} c_{i} (G) ⟨ x ⟩_{i},$ where $ \lan…

Normalized skew Schur polynomials are Lorentzian

Philip B. Zhang

We prove the conjecture of Huh, Matherne, Mészáros, and St.~Dizier that the normalization of every skew Schur polynomial in finitely many variables is Lorentzian. We first realize…

Real-rootedness of Kazhdan--Lusztig and $Z$ -polynomials of thagomizer matroids and graphic matroids of $K_{2, n}$

Philip B. Zhang

Let $T_{n} = K_{1, 1, n}$ , and let $P_{n} (x)$ denote the Kazhdan--Lusztig polynomial of its graphic matroid. We prove that, whenever $n \geq 2$ and $0 \leq λ \leq n /2$ , the polynomial $P_{n} (x) + λ x$ …

Simultaneous avoidance of length-4 patterns in ascent sequences

Qi Liu, Sergey Kitaev, Philip B. Zhang

Ascent sequences form a central class of combinatorial objects, as they are in bijection with several important families such as (2+2)-free posets, Stoimenow matchings, and other F…

Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action

Matthew H. Y. Xie, Philip B. Zhang, Michael X. X. Zhong

The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph $K_{1, 1, n}$ , has full automorphism group isomorphic to the hyperoctahedral group whenever $…

Catalan structures arising from pattern-avoiding Stoimenow matchings and other Fishburn objects

Shuzhen Lv, Sergey Kitaev, Philip B. Zhang

In connection with Vassiliev's knot invariants, Stoimenow introduced in 1998 a class of matchings, also known as regular linearized chord diagrams. These matchings are linked to va…

Joint equidistributions of mesh patterns 123 and 132 with minus antipodal shadings

Shuzhen Lv, Philip B. Zhang

The study of joint equidistributions of mesh patterns 123 and 132 with the same symmetric shadings was recently initiated by Kitaev and Lv, where 75 of 80 potential joint equidistr…

Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids

Matthew H. Y. Xie, Philip B. Zhang

Gao and Xie (2021) conjectured that the inverse Kazhdan-Lusztig polynomial of any matroid is log-concave. Although the inverse Kazhdan-Lusztig polynomial may not always have only r…

Descent generating polynomials for ( $n - 3$ )- and ( $n - 4$ )-stack-sortable (pattern-avoiding) permutations

Sergey Kitaev, Philip B. Zhang

In this paper, we find distribution of descents over $(n - 3)$ - and $(n - 4)$ -stack-sortable permutations in terms of Eulerian polynomials. Our results generalize the enumeration resul…

Joint equidistributions of mesh patterns 123 and 321 with symmetric and minus-antipodal shadings

Shuzhen Lv, Philip B. Zhang

In this paper, we extend recent results by Lv and Kitaev by proving 20 (out of 22 possible) joint equidistributions of mesh patterns 123 and 321 with symmetric shadings, as well as…

Distributions of mesh patterns of short lengths on king permutations

Dan Li, Philip B. Zhang

Brändén and Claesson introduced the concept of mesh patterns in 2011, and since then, these patterns have attracted significant attention in the literature. Subsequently, in 2015,…

math.CO2024★ 4 cited

Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs

Tian Han, Sergey Kitaev, Philip B. Zhang

In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations…

On (shape-)Wilf-equivalence of certain sets of (partially ordered) patterns

Alexander Burstein, Tian Han, Sergey Kitaev +1

We prove a conjecture of Gao and Kitaev on Wilf-equivalence of sets of patterns {12345,12354} and {45123,45213} that extends the list of 10 related conjectures proved in the litera…

Distributions of statistics on separable permutations

Joanna N. Chen, Sergey Kitaev, Philip B. Zhang

We derive functional equations for distributions of six classical statistics (ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, and right-to-left…