76 problems

Let $e_n$ be the elementary symmetric function, let $\Delta'_{e_{k-1}}$ be the modified Macdonald eigenoperator, and let $\mathcal{LD}_n$ be the set of labeled Dyck paths of size…

Chapoton's conjecture. These polynomials are related by the invertible transformation

Let $C_n(q,t)$ be the $q,t$-analog of the Catalan numbers, and let $\operatorname{area}(\pi)$ and $\operatorname{bounce}(\pi)$ denote Haglund's area and bounce statistics on Dyck p…

Let $n,k\in\mathbb{N}$ with $n>k\geq 0$. For a labelled Dyck path $P$ of size $n$ with positive labels and $k$ decorated rises, let $\mathsf{area}(P)$, $\mathsf{dinv}(P)$, and…

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

Let $a$ and $b$ be two co-prime integers. Let $\mathcal{L}_{a,b}$ be the poset of lattice paths in the rectangle $a\times b$, and let $\mathrm{Dyck}_{a,b}$ be the poset of $(a,b)$-…

Let $\mathcal{LD}_n$ be the set of labeled Dyck paths of size $n$. For $P\in\mathcal{LD}_n$, let $a_i(P)$, $\ell_i(P)$, $\operatorname{area}(P)$, $\operatorname{dinv}(P)$, $d_i(P)$…

Let $u$ and $v$ be Dyck paths of lengths $L$ and $L+1$, respectively. Let $^*v$ be the Ballot path of length $L$ obtained by left reduction of $v$, so that … Let ${\cal O}_l(w,s)$…

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 $n$ be a positive integer, and consider the Dyck random transpositions walk on Catalan strings of order $n$, equivalently on Dyck paths of length $2n$. The chain chooses two in…

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…

Let $C_n^{(k)}(q,t)=\left<\nabla^{k}e_n,e_n\right>$ denote the higher $q,t$-Catalan polynomial, and let $k$-Dyck paths be the associated generalized Dyck paths, with area and bounc…

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 $\Theta_{e_k}$ be the Theta operator, let $\nabla$ be the nabla operator, and let $\mathsf{LD}(n)^{\ast k}$ and $\mathsf{LD}(n)^{\bullet k}$ denote the rise- and valley-decorat…

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…

Let $\mathcal{S}_n=(\mathcal{D}_n,\leq)$ be the lattice whose elements are Dyck paths of semilength $n$, ordered by the relation defined in the paper. The diameter of…

Lin's restricted bijection conjecture. The restriction of $\Phi$ to $\mathcal{F}(\mathcal{D}_n)$ is a bijection

Lin's bijection conjecture. The map $\Phi$ is a bijection between the desired sets; in particular, $\Phi(\beta)$ is a compatible pair for every…