54 problems

Let $D_n^{(r)}$ be the set of all $r$-fans of Dyck paths of length $2n$, and define … where $[m]_q=1+q+q^2+\cdots+q^{m-1}$. Let $C_{2n}$ be the cyclic group of order $2n$ acting on…

Let $\mathcal U_n$ be the set of unit interval orders on $n$ elements, and let $\mathcal D_n$ be the set of Dyck paths from $(0,0)$ to $(n,n)$. For $U\in\mathcal U_n$, let $a(U)$ b…

Fix $n\ge1$, a diagonal multiset $m$ realizable by an area word of size $n$, $k\ge0$, $r\ge1$, a scaffold $\Xi$, and a matrix $g=(g_{t,\delta})$ recording per-gap, per-diagonal num…

Scaffold exchange conjecture. For every $n\ge1$, $k\ge0$, $r\ge1$, every diagonal multiset $m$ of size $n$, and every scaffold $\Xi$,

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(q,t)$, let $\{\widetilde H_\mu\}$ be the modified Macdonald basis, and define $\Delta'_f$ by … where…

Let $\vec{k}$ be a vector of positive integers and define the area-bounce polynomial … A polynomial is $q,t$-symmetric when $F(q,t)=F(t,q)$. The $(a,b,\ldots,b)$-symmetry conjectur…

Let $\vec{k}$ be a vector of positive integers, let $\mathcal{K}$ be the associated parameter, and define the area-bounce polynomial … Write $\lambda(\mathcal{K})$ for the partitio…

Let $\vec{k}$ be a vector of positive integers, let $\mathcal{K}$ denote the corresponding parameter indexing the relevant $\vec{k}$-Dyck paths, and let…

Reflection-symmetric strengthening. There exists a set $\Pi=\{\pi_1,\pi_2,\ldots,\pi_{C_k}\}$ of $C_k$ permutations of $\mathbb{Z}_{2k+1}$, each fixing $0$, and a bijection…

Dyck-path strengthening. There exists a set $\Pi=\{\pi_1,\pi_2,\ldots,\pi_{C_k}\}$ of $C_k$ permutations of $\mathbb{Z}_{2k+1}$, each fixing $0$, and a bijection…

Let $W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^m$, where $w_{n,k,m}$ counts Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. For real-rooted polyno…

Let $W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^m$, where $w_{n,k,m}$ counts Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. A sequence of real-roo…

Hawkes's degreewise conjecture. For every $d$,

Hawkes's conjecture. The rational $q,t$-Catalan polynomial satisfies

Let $P$ be a unit interval order corresponding to an incomparability graph $G(d)$ and to a lex-maximal part listing encoded by ${\bf a}=(a_1,\ldots,a_n)$. Let $d'$ be the Dyck path…

Let $d$ be a Dyck path, let $G(d)$ be its indifference graph, and let $X_{G(d)}$ be its chromatic symmetric function. For any finite number of variables, restrict $X_{G(d)}$ to tho…

Let $d$ be a Dyck path, let $G(d)$ be its indifference graph, and let $X_{G(d)}$ be its chromatic symmetric function. For any finite number of variables, restrict $X_{G(d)}$ to tho…

Extremal-set periodicity conjecture. The sequence $H(D^{(m,V)})$ is periodic and has the form

Let $n$ be a positive integer. An up-down permutation of size $2n$ is a permutation of $\{1,\ldots,2n\}$ whose successive entries alternately rise and fall, and a permutation avoid…

Let $m,n,k$ be in the range for which $\mathsf{PLD}(m,n)^{\ast k}$ is defined, and let $\mathsf{pmaj}(D)$ denote the proposed partial-major-index statistic on a partially labelled…

Let $m,n,k$ be in the range for which the partially labelled Dyck-path set $\mathsf{PLD}(m,n)^{\ast k}$ is defined. Let $\Delta_{h_m}$ and $\Delta'_{e_{n-k-1}}$ be Delta operators,…

Alternating Möbius function conjecture. The Möbius function is alternating, meaning that it is

Rank-3 maximum conjecture. The maximum absolute value of the Möbius function on intervals of rank $3$ is

Differential-equation conjecture. The generating function $D(s,z)$ is D-finite in $z$ but not algebraic. Specifically, it satisfies a fifth-order linear differential equation with…

Let $\rho=(a\,|\,b)$ be a hook, let $\Gamma_a$ be the Dyck path defined in the source, and let $\mathbb{L}_\beta(t;\bm{z})$ be the LLT polynomial associated with a Dyck path…