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Kathy Q. Ji

Publications and source records attributed to Kathy Q. Ji.

At least 19 recordsLinked to original sources
arXiv2026

Revisiting $q$-Derangement Numbers via Decorated Permutations

This paper aims to provide a direct combinatorial proof of the Gessel-Wachs formula for $q$-derangement numbers in the setting of decorated permutations, without using the $q$-binomial inversion formula. Decorated permutations, introduced by Postnikov in his study of the totally nonnegative Grassmannian, provide a natural framework for Chen's signed fixed-point model. Our proof is based on a major-index generating function for decorated permutations with a fixed number of signed fixed points, together with a sign-reversing and descent-set-preserving involution, thereby answering a question raised by Chen. This involution was discovered through human--AI collaboration. Moreover, our framework yields an immediate proof of a result of Désarménien and Wachs concerning the equidistribution between descent classes of derangements and descent classes of desarrangements. We also supply combinatorial proofs of two recurrence relations for the $q$-derangement numbers in the setting of desarrangements with the aid of the insertion lemma for ordinary permutations.

math.CO↗
arXiv2026

Labeled Plane Trees and Increasing Plane Trees

The main aim of this paper is to establish a polynomial analogue of $(n+1)!C_n=2^n(2n-1)!!$ (with $C_n$ as the $n$-th Catalan number) in the setting of labeled plane trees and increasing plane trees. This analogue is formulated in terms of improper edges of labeled plane trees and yields explicit formulas for the generating polynomials defined on labeled plane trees refined by improper and proper edges, together with a root-degree refinement for trees rooted at $0$. To prove this result, we construct a new involution on labeled plane trees, which implies that the number of improper edges and the number of proper edges are equidistributed over the set of labeled plane trees. We further apply this involution to establish pairwise symmetry properties of multivariable polynomials defined on labeled plane trees involving several classes of leaves and interior vertices. More precisely, certain specializations of these polynomials are invariant under the subgroup of $S_6$ generated by the three disjoint transpositions $(12)$, $(34)$, and $(56)$. As special cases, our results recover the symmetry properties for plane trees and tip-augmented plane trees due to Dong, Du, Ji and Zhang. Finally, via the Koganov--Janson correspondence, improper edges of labeled plane trees correspond bijectively to improper arcs of quasi-Stirling permutations, leading to an explicit formula for the generating function defined on quasi-Stirling permutations refined by improper arcs.

math.CO↗
arXiv2026

$q$-Derivative Grammar

Context-free grammars, originating in computer science, are related to enumerative combinatorics through two distinct lines of development pioneered by Schützenberger and Chen, respectively. In the framework established by Schützenberger and Delest-Schützenberger-Viennot, unambiguous grammars are translated into functional equations for ordinary generating functions. Inspired by Rota's umbral calculus, Chen later developed a grammatical calculus by associating each context-free grammar with a formal derivative operator. Dumont further developed this method through numerous combinatorial interpretations of grammars with finite and infinite alphabets. Substantial progress in this direction has been achieved over the last decade. In this paper, we introduce a q-analogue of grammatical calculus, which we call the q-derivative grammar. We establish the basic framework of q-grammars and develop the q-grammatical calculus for computing q-exponential generating functions associated with q-grammars. Concrete q-grammars are constructed to study q-Eulerian, q-Roselle and q-André polynomials, including their generating functions and recurrences. This work extends the grammatical method to the q-setting and opens up new research directions.

math.CO↗
arXiv2026

On the distributions of the statistics (des, maj, inv) over several classes of permutations

We investigate the joint distribution of the trivariate statistics (des, maj, inv) on classical permutations, Andre permutations of the first and second kinds, and Simsun permutations. By decomposing permutations according to the position of the smallest element, we obtain explicit recurrence relations for the generating functions of these statistics. In the classical permutation setting, our recurrence relation yields the generating function for the trivariate statistics (des, maj, inv) due to Gessel, which is typically proved using MacMahon's technique.

math.CO↗
arXiv2026

Modular Nahm sums for symmetrizable matrices of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$

In this paper, we present three families of modular Nahm sums for symmetrizable matrices with arbitrary rank $r\geq 2$ of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$. Specifically, the cases corresponding to $r = 2$ and $r = 3$ of these families have been previously demonstrated by Mizuno, Warnaar, and B. Wang-L. Wang. Building upon these three families, we construct two vector-valued automorphic forms, one of which is a vector-valued modular function when $r$ is odd.

math.CO↗
arXiv2025

Inverse descent statistic for André and simsun permutations

Simsun permutations, André I permutations and André II permutations are three combinatorial models for Euler numbers. It's known that the descent statistic is equidistributed over the set of André I permutations and the set of simsun permutations. In this paper, we prove that the trivariate statistic (ides, des, maj), comprising the inverse descent, descent, and major index, are equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of André I permutations and the set of André II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem.

math.CO↗
arXiv2024

Signed Mahonian Polynomials on Derangements in Classical Weyl Groups

The polynomial of the major index ${\rm maj}_W (σ)$ over the subset $T$ of the Coxeter group $W$ is called the Mahonian polynomial over $T$, where ${\rm maj}_W (σ)$ is a Mahonian statistic of an element $σ\in T$, whereas the polynomial of the major index ${\rm maj}_W (σ)$ with the sign $(-1)^{\ell_W(σ)}$ over the subset $T$ is referred to as the signed Mahonian polynomial over $T$, where ${\ell_W(σ)}$ is the length of $σ\in T$. Gessel, Wachs, and Chow established the formulas for the Mahonian polynomials over the sets of derangements in the symmetric group $S_n$ and the hyperoctahedral group $B_n$. By extending Wachs' approach and employing a refinement of Stanley's shuffle theorem established in our recent paper, we derive the formula for the Mahonian polynomials over the set of derangements in the even-signed permutation group $D_n$. This completes a picture which is now known for all the classical Weyl groups. Gessel-Simion, Adin-Gessel-Roichman, and Biagioli previously established formulas for the signed Mahonian polynomials over the classical Weyl groups. Building upon their formulas, we derive the formulas for the signed Mahonian polynomials over the set of derangements in classical Weyl groups. As applications of the formulas for the (signed) Mahonian polynomials over the sets of derangements in the classical Weyl groups, we obtain enumerative formulas of the number of derangements in classical Weyl groups with even lengths.

math.CO↗
arXiv2024

New refinements of Narayana polynomials and Motzkin polynomials

Chen, Deutsch and Elizalde introduced a refinement of the Narayana polynomials by distinguishing between old (leftmost child) and young leaves of plane trees. They also provided a refinement of Coker's formula by constructing a bijection. In fact, Coker's formula establishes a connection between the Narayana polynomials and the Motzkin polynomials, which implies the $γ$-positivity of the Narayana polynomials. In this paper, we introduce the polynomial $G_{n}(x_{11},x_{12},x_2;y_{11},y_{12},y_2)$, which further refine the Narayana polynomials by considering leaves of plane trees that have no siblings. We obtain the generating function for $G_n(x_{11},x_{12},x_2;y_{11},y_{12},y_2)$. To achieve further refinement of Coker's formula based on the polynomial $G_n(x_{11},x_{12},x_2;y_{11},y_{12},y_2)$, we consider a refinement $M_n(u_1,u_2,u_3;v_1,v_2)$ of the Motzkin polynomials by classifying the old leaves of a tip-augmented plane tree into three categories and the young leaves into two categories. The generating function for $M_n(u_1,u_2,u_3;v_1,v_2)$ is also established, and the refinement of Coker's formula is immediately derived by combining the generating function for $G_n(x_{11},x_{12},x_2;y_{11},y_{12},y_2)$ and the generating function for $M_n(u_1,u_2,u_3;v_1,v_2)$. We derive several interesting consequences from this refinement of Coker's formula. The method used in this paper is the grammatical approach introduced by Chen. We develop a unified grammatical approach to exploring polynomials associated with the statistics defined on plane trees. As you will see, the derivations of the generating functions for $G_n(x_{11},x_{12},x_2;{y}_{11},{y}_{12},y_2)$ and $M_n(u_1,u_2,u_3;v_1,v_2)$ become quite simple once their grammars are established.

math.CO↗
arXiv2024

A Refinement of a Theorem of Diaconis-Evans-Graham

The note is dedicated to refining a theorem by Diaconis, Evans, and Graham concerning successions and fixed points of permutations. This refinement specifically addresses non-adjacent successions, predecessors, excedances, and drops of permutations.

math.CO↗
arXiv2024

Overpartitions and Bressoud's conjecture, II

The main objective of this paper is to present an answer to Bressoud's conjecture for the case $j=0$, resulting in a complete solution to the conjecture. The case for $j=1$ has been recently resolved by Kim. Using the connection established in our previous paper between the ordinary partition function $B_0$ and the overpartition function $\overline{B}_1$, we found that the proof of Bressoud's conjecture for the case $j=0$ is equivalent to establishing an overpartition analogue of the conjecture for $j=1$. By generalizing Kim's method, we obtain the desired overpartition analogue of Bressoud's conjecture for $j=1$, which eventually enables us to confirm Bressoud's conjecture for the case $j=0$.

math.CO↗
arXiv2023

Some Refinements of Stanley's Shuffle Theorem

We first give a combinatorial proof of Stanley's shuffle theorem by using the insertion lemma of Haglund, Loehr and Remmel. Based on this combinatorial construction, we establish several refinements of Stanley's shuffle theorem.

math.CO↗
arXiv2023

Convexity and log-concavity of the partition function weighted by the parity of the crank

Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (reps. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound, we show that $M_k(n-1)+M_k(n+1)>2M_k(n)$ for $k=0$ or $1$ and $n\geq 39$. This result can be seen as the refinement of the classical result regarding the convexity of the partition function $p(n)$, which counts the number of partitions of $n$. We also show that $M_0(n)$ (resp. $M_1(n)$) is log-concave for $n\geq 94$ and satisfies the higher order Turán inequalities for $n\geq 207$ with the aid of the upper bound and the lower bound for $M_0(n)$ and $M_1(n)$.

math.CO↗
arXiv2023

The Binomial-Stirling-Eulerian Polynomials

We introduce the binomial-Stirling-Eulerian polynomials, denoted $\tilde{A}_n(x,y|α)$, which encompass binomial coefficients, Eulerian numbers and two Stirling statistics: the left-to-right minima and the right-to-left minima. When $α=1$, these polynomials reduce to the binomial-Eulerian polynomials $\tilde{A}_n(x,y)$, originally named by Shareshian and Wachs and explored by Chung-Graham-Knuth and Postnikov-Reiner-Williams. We investigate the $γ$-positivity of $\tilde{A}_n(x,y|α)$ from two aspects: firstly by employing the grammatical calculus introduced by Chen; and secondly by constructing a new group action on permutations. These results extend the symmetric Eulerian identity found by Chung, Graham and Knuth, and the $γ$-positivity of $\tilde{A}_n(x,y)$ first demonstrated by Postnikov, Reiner and Williams.

math.CO↗
arXiv2023

The $(α,β)$-Eulerian Polynomials and Descent-Stirling Statistics on Permutations

Carlitz and Scoville introduced the polynomials $A_n(x,y|α,β)$, which we refer to as the $(α, β)$-Eulerian polynomials. These polynomials count permutations based on Eulerian-Stirling statistics, including descents, ascents, left-to-right maxima, and right-to-left maxima. Carlitz and Scoville obtained the generating function of $A_n(x,y|α,β)$. In this paper, we introduce a new family of polynomials, $P_n(u_1,u_2,u_3,u_4|α,β)$, defined on permutations, incorporating descent-Stirling statistics including valleys, exterior peaks, right double descents, left double ascents, left-to-right maxima, and right-to-left maxima. By employing the grammatical calculus introduced by Chen, we establish the connection between the generating function of $P_n(u_1,u_2,u_3,u_4|α,β)$ and the generating function of the $(α,β)$-Eulerian polynomials $A_n(x,y|α,β)$ introduced by Carlitz and Scoville. Using this connection, we derive the generating function of $P_n(u_1,u_2,u_3,u_4|α,β)$, which can be specialized to obtain the $(α,β)$-extensions of generating functions for peaks, left peaks, double ascents, right double ascents and left-right double ascents given by David-Barton, Elizalde and Noy, Entringer, Gessel, Kitaev and Zhuang. Moreover, we establish two relations between $P_n(u_1,u_2,u_3,u_4|α,β)$ and $A_n(x,y|α,β)$, which enable us to derive $(α,β)$-extensions of results of Stembridge, Petersen, Brändén, and Zhuang. Specializing $(α,β)$-extensions of Stembridge's formula and the left peak version of Stembridge's formula allows us to derive the $(α,β)$-extensions of the tangent and secant numbers.

math.CO↗
arXiv2023

Unimodality of $k$-Regular Partitions into Distinct Parts with Bounded Largest Part

A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the largest part is at most $km+k-1$. Let $d_{k,m}(n)$ denote the number of $k$-regular partition of $n$ into distinct parts where the largest part is at most $km+k-1$. In line with this method, we show that $d_{4,m}(n)\geq d_{4,m}(n-1)$ for $m\geq 0$, $1\leq n\leq 3(m+1)^2$ and $n\neq 4$ and $d_{8,m}(n)\geq d_{8,m}(n-1)$ for $m\geq 2$ and $1\leq n\leq 14(m+1)^2$. When $5\leq k\leq 10$ and $k\neq 8$, we show that $d_{k,m}(n)\geq d_{k,m}(n-1)$ for $m\geq 0$ and $1\leq n\leq \left\lfloor\frac{k(k-1)(m+1)^2}{4}\right\rfloor$.

math.CO↗
arXiv2023

Higher Order Turan Inequalities for the Distinct Partition Function

We prove that the number $q(n)$ of partitions into distinct parts is log-concave for $n \geq 33$ and satisfies the higher order Turán inequalities for $n\geq 121$ conjectured by Craig and Pun. In doing so, we establish explicit error terms for $q(n)$ and for $q(n-1)q(n+1)/q(n)^2$ based on Chern's asymptotic formulas for $η$-quotients.

math.CO↗