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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.

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., Comment: To appear in Discrete Applied Mathematics

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., Comment: 23 pages

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 $\Delta'_{e_k} e_n$. Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression $\Delta'_{e_k} \Delta_{h_r}e_n$, 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 $\Delta'_{e_k} \Delta_{h_r}e_n$ are equivalent when $t$ or $q$ equals 0, thus proving the valley version of the Extended Delta Conjecture when $t$ or $q$ equals 0., Comment: 28 pages, 9 figures

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., Comment: 19 pages, 15 figures

Classical pattern avoidance and occurrence are well studied in the symmetric group $\mathcal{S}_{n}$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in the set of 132-avoiding permutations and the set of 123-avoiding permutations., Comment: 23 pages, 5 fugures

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 $\gamma$-matches in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$ for each length 3 consecutive pattern $\gamma$. 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., Comment: 31 pages, 10 figures

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 $3$., Comment: 35 pages, 18 figures

An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined $\mathcal{OP}_{n,k}(\sigma)$ to be the set of ordered set partitions in $\mathcal{OP}_{n,k}$ avoiding a permutation pattern $\sigma$ and obtained the formula for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $3$. In this paper, we define a new pattern avoidance for the set $\mathcal{OP}_{n,k}$, called $\mathcal{WOP}_{n,k}(\sigma)$, which includes the questions proposed by Godbole, Goyt, Herdan and Pudwell. We obtain formulas for $|\mathcal{WOP}_{n,k}(\sigma)|$ combinatorially for any $\sigma$ of length $ 3$. We also define 3 kinds of descent statistics on ordered set partitions and study the distribution of the descent statistics on $\mathcal{WOP}_{n,k}(\sigma)$ for $\sigma$ of length $3$., Comment: 42 pages, 16 figures

Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case $k=2$; We show that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_2} e_n[X]$ has a positive expansion in terms of $q,t$-analogs.

Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions.