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Dun Qiu

Publications and source records attributed to Dun Qiu.

At least 19 recordsLinked to original sources
arXiv2026

Entry and position statistics in alternating permutations and Young tableaux

We study entry and position statistics in alternating permutations and standard Young tableaux using linear extensions and generating functions. For alternating permutations, we give explicit formulas for the mean and variance of the first entry and of the position of any fixed value. For standard Young tableaux, we express the sum of the lower-left entries as the number of tableaux of an associated skew shape. For two-row shapes, we determine the distribution of the entry in each cell of the second row, and for the shape $(n,n)$ we give a closed formula for its sum over all tableaux, which recovers a limit formula of Stanley. Through a bijection of Lewis, the results for the shape $(n,n)$ describe the entries of a random $321$-avoiding down-up permutation of even length.

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arXiv2026

The Schur positivity of Lucas binomial differences

For positive integers $a\le b\le c\le d$ with $ad=bc$, we prove that the Lucas binomial coefficients satisfy $(-1)^{a+1}(L_{b,c}[X]-L_{a,d}[X])\in\mathbb{N}[e_1[X],e_2[X],h_2[X]]$, resolving Bergeron's conjecture on Schur positivity in two variables. We also prove the same positivity for $L_{u,v}[X]-e_1[X]^{uv-rs}L_{r,s}[X]$ when $rs<uv$. Using Lucas atoms, we establish $e$-positivity for degree-adjusted Lucas multinomial differences under dominance order.

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arXiv2026

The $(km,kn)$-square paths theorem

For positive integers $k,m,n$ with $\gcd(m,n)=1$, we prove a $(km,kn)$-square paths theorem: the symmetric function $(-1)^{k-1}\frac{[km]_q}{[k]_q}p_k[-MX^{m,n}]\cdot1$ is a weighted sum over $(km,kn)$-preference functions. This proves a corrected form of the conjecture of Iraci, Pagaria, Paolini, and Vanden Wyngaerd and generalizes the square paths theorem of Sergel. We encode preference functions by row strict tableaux and closed walks on a set of states, compare them with parking functions for each value of the touch statistic, and apply Mellit's compositional $(km,kn)$-shuffle theorem. Finally, we establish a fall Delta square theorem conjectured by Iraci, Pagaria, and Paolini.

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arXiv2026

The nabla operator, Hall--Littlewood polynomials and LLT polynomials

Bergeron, Garsia, Haiman, and Tesler conjectured that $(-1)^{n-\ell(μ)}\nabla\widetilde H_μ[X;0,t]$ and $\nablaω\widetilde H_μ[X;0,t^{-1}]$ are Schur positive, and that $\nablaω\widetilde H_μ[X;0,t^{-1}]-\nablaω\widetilde H_ν[X;0,t^{-1}]$ is Schur positive when $μ$ dominates $ν$. The first statement follows from the compositional shuffle conjecture of Haglund, Morse, and Zabrocki, proved by Carlsson and Mellit. We prove the other two statements. All three statements also hold for $\nabla^r$ with $r\geq1$. The main new step is an identity: when $μ$ covers $ν$, the difference $ω\widetilde H_μ[X;0,t^{-1}]-ω\widetilde H_ν[X;0,t^{-1}]$ is minus a power of $t$ times one LLT polynomial. Using the Catalanimals and the Cauchy identity of Blasiak, Haiman, Morse, Pun, and Seelinger, we show that $\nabla^r$ of this LLT polynomial is, up to sign, a positive sum of LLT polynomials. We also extend the main results to the Schiffmann algebra setting.

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arXiv2026

Coincidences and Growth of Boxed Mesh Patterns

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 $2^{5n}n^{βn}$, where $β=\log_2(2\cos(π/7))<0.85$. 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 $n\log n$. For Box(12), we identify the occurrence statistic with the up-degree in the strong Bruhat order, obtaining its maximum, its mean, and an exact insertion identity for the distribution polynomials. We conjecture that the coefficients of these polynomials are unimodal.

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arXiv2026

The Ehrhart series of magic squares of orders seven and eight

Let $\mathrm{IMS}_n(m)$ count the $n\times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$. We determine the Ehrhart series $F_n(q)=\sum_{m\geq0}\mathrm{IMS}_n(m)q^m$ as reduced rational functions for $n=7$ and $n=8$. Their numerator--denominator degrees are respectively $(366,373)$ and $(540,548)$. Both numerators have positive integer coefficients, are palindromic and strictly unimodal. The proofs share one finite architecture: a signed SimpCone decomposition is evaluated by quotient characters over finite fields, a certified common denominator and Ehrhart reciprocity reduce the rational identity to finitely many coefficients, an explicit counting bound lifts modular congruences to integer equalities, and exact gcd computations prove reducedness. For order eight, a face-index pole certificate gives a degree-$598$ common denominator without enumerating the full face lattice, leaving $296$ independent coefficients in degrees $0$ through $295$. Once the candidate rational function is known, the first six production primes certify this finite prefix by the same bounded-coefficient argument used for order seven.

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arXiv2026

The Schur positivity of $\nabla m_μ$

Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $μ,λ\vdash n$, the polynomial $(-1)^{|μ|-\ell(μ)}\langle \nabla m_μ, s_λ\rangle$ has nonnegative integer coefficients, where $\nabla$ is the Bergeron--Garsia nabla operator, which acts diagonally on the modified Macdonald basis, and $m_μ$ is the monomial symmetric function. In this article, we prove this conjecture, and more generally that $(-1)^{|μ|-\ell(μ)}\langle\nabla^r m_μ,s_λ\rangle\in\mathbb{N}[q,t]$ for all $r\geq 1$. We establish a recursion showing that $(-1)^{|μ|-\ell(μ)}m_μ$ has an expansion with coefficients in $\mathbb{Q}_{\geq 0}[q]$ in the symmetric functions $C_α(1)$, where $C_a$ denotes the operator introduced by Haglund, Morse and Zabrocki. Combining this expansion with the compositional shuffle theorems of Carlsson--Mellit and Mellit, and with the Schur positivity of LLT polynomials, completes the proof. The same method, using the $e$-positivity of column LLT polynomials after the substitution $q\mapsto q+1$, also gives an $e$-positive analogue.

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arXiv2026

The nonsymmetric compositional Delta theorem

Extending the symmetric framework of D'Adderio and Mellit, we establish a nonsymmetric generalization of the compositional Delta theorem. Building on Blasiak et al.'s theory of flagged LLT polynomials, we derive signed and unsigned nonsymmetric identities evaluated in terms of flagged LLT polynomials. Furthermore, by introducing nonsymmetric variants of the $\nabla$ and $τ^*$ operators, we obtain a novel operator formulation. We show that applying Weyl symmetrization to these nonsymmetric identities systematically recovers the original compositional Delta theorem. Finally, we propose analogous conjectures regarding stable atom positivity.

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arXiv2026

On shortening universal words for multi-dimensional permutations

A universal word (u-word) for $d$-dimensional permutations of length $n$ is a 2-dimensional word with $d-1$ rows, any size $n$ window of which is order-isomorphic to exactly one permutation of length $n$, and all permutations of length $n$ are covered. It is known that u-words (in fact, even u-cycles, a stronger claim) for $d$-dimensional permutations exist. In this paper, we use the idea of incomparable elements to prove that u-words of length $(n!)^{d-1}+n-1-i(n-1)$, for $d\geq 2$ and $$0\leq i\leq \frac{2^{d-1}}{n-1}\left[(1+(n-1)!)^{d-1}-\left(1+\frac{(n-1)!}{2}\right)^{d-1}\right],$$ for $d$-dimensional permutations of length $n$ exist, which generalizes the respective result of Kitaev, Potapov and Vajnovszki for ``usual'' permutations ($d=2$).

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arXiv2026

Symmetric statistics on rational Dyck paths

Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run and ratio-run are introduced. They both have symmetric joint distributions with the return statistic. We give combinatorial proofs and algebraic proofs of the symmetries, generalizing a result of Li and Lin.

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arXiv2024

Stanley's conjecture on the Schur positivity of distributive lattices

In this paper we solve an open problem on distributive lattices, which was proposed by Stanley in 1998. This problem was motivated by a conjecture due to Griggs, which equivalently states that the incomparability graph of the boolean algebra $B_n$ is nice. Stanley introduced the idea of studying the nice property of a graph by investigating the Schur positivity of its corresponding chromatic symmetric functions. Since the boolean algebras form a special class of distributive lattices, Stanley raised the question of whether the incomparability graph of any distributive lattice is Schur positive. Stanley further noted that this seems quite unlikely. In this paper, we construct a family of distributive lattices which are not nice and hence not Schur positive. We also provide a family of distributive lattices which are nice but not Schur positive.

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arXiv2024

On a family of universal cycles for multi-dimensional permutations

A universal cycle (u-cycle) for permutations of length $n$ is a cyclic word, any size $n$ window of which is order-isomorphic to exactly one permutation of length $n$, and all permutations of length $n$ are covered. It is known that u-cycles for permutations exist, and they have been considered in the literature in several papers from different points of view. In this paper, we show how to construct a family of u-cycles for multi-dimensional permutations, which is based on applying an appropriate greedy algorithm. Our construction is a generalisation of the greedy way by Gao et al. to construct u-cycles for permutations. We also note the existence of u-cycles for $d$-dimensional matrices.

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arXiv2022

Pattern-avoidance and Fuss-Catalan numbers

We study a subset of permutations, where entries are restricted to having the same remainder as the index, modulo some integer $k \geq 2$. We show that when also imposing the classical 132- or 213-avoidance restriction on the permutations, we recover the Fuss--Catalan numbers and some special cases of the Raney numbers. Surprisingly, an analogous statement also holds when we impose the mod $k$ restriction on a Catalan family of subexcedant functions. Finally, we completely enumerate all combinations of mod-$k$-alternating permutations, avoiding two patterns of length 3. This is analogous to the systematic study by Simion and Schmidt, of permutations avoiding two patterns of length 3.

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arXiv2020

Schur Function Expansions and the Rational Shuffle Theorem

Gorsky and Negut introduced operators $Q_{m,n}$ on symmetric functions and conjectured that, in the case where $m$ and $n$ are relatively prime, the expression ${Q}_{m,n}(1)$ is given by the Hikita polynomial ${H}_{m,n}[X;q,t]$. Later, Bergeron-Garsia-Leven-Xin extended and refined the conjectures of ${Q}_{m,n}(1)$ for arbitrary $m$ and $n$ which we call the Extended Rational Shuffle Conjecture. In the special case ${Q}_{n+1,n}(1)$, the Rational Shuffle Conjecture becomes the Shuffle Conjecture of Haglund-Haiman-Loehr-Remmel-Ulyanov, which was proved in 2015 by Carlsson and Mellit as the Shuffle Theorem. The Extended Rational Shuffle Conjecture was later proved by Mellit as the Extended Rational Shuffle Theorem. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of ${Q}_{m,n}(1)$ in certain special cases. Leven gave a combinatorial proof of the Schur function expansion of ${Q}_{2,2n+1}(1)$ and ${Q}_{2n+1,2}(1)$. In this paper, we explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of ${Q}_{m,n}(1)$. Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of ${Q}_{m,n}(1)$ in the case where $m$ or $n$ equals $3$.

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arXiv2019

The valley version of the Extended Delta Conjecture

The Shuffle Theorem of Carlsson and Mellit gives a combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics, and the Delta Conjecture of Haglund, Remmel and the second author provides two generalizations of the Shuffle Theorem to the delta operator expression $Δ'_{e_k} e_n$. Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression $Δ'_{e_k} Δ_{h_r}e_n$, which is analogous to the rise version of the Delta Conjecture. Recently, D'Adderio, Iraci and Wyngaerd proved the rise version of the Extended Delta Conjecture at the case when $t=0$. In this paper, we propose a new valley version of the Extended Delta Conjecture. Then, we work on the combinatorics of extended ordered multiset partitions to prove that the two conjectures for $Δ'_{e_k} Δ_{h_r}e_n$ are equivalent when $t$ or $q$ equals 0, thus proving the valley version of the Extended Delta Conjecture when $t$ or $q$ equals 0.

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arXiv2019

$e$-Positivity Results and Conjectures

In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions $G[X;q]$ with the property that $G[X;1+q]$ is $e$-positive. A large subvariety of his examples could be explained by the conjecture that the Dyck path LLT polynomials exhibit the same phenomenon. In this paper we list the results of computer explorations which suggest that other examples exhibit the same phenomenon. We prove two of the resulting conjectures and propose algorithms that would prove several of our conjectures. In writing this paper we have learned that similar findings have been independently discovered by Per Alexandersson.

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arXiv2019

Counting Consecutive Pattern Matches in $\mathcal{S}_n(132)$ and $\mathcal{S}_n(123)$

In this paper, we study the distribution of consecutive patterns in the set of 123-avoiding permutations and the set of 132-avoiding permutations, that is, in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$. We first study the distribution of consecutive pattern $γ$-matches in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$ for each length 3 consecutive pattern $γ$. Then we extend our methods to study the joint distributions of multiple consecutive patterns. Some more general cases are discussed in this paper as well.

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