33 problems

Given coprime positive integers $m$ and $n$, let $\mathcal D_{m,n}$ be the set of rational $(m,n)$-Dyck paths, and let $\Phi$ be the sweep map that sorts the steps of a path accord…

For $k\leqslant n$, a left Gog or GOGAm trapezoid of shape $(n,k)$ is obtained from the corresponding triangle by retaining the $k$ leftmost diagonals. Left Gog–GOGAm trapezoid con…

A quartic Eulerian orientation without alternating vertices is a quartic Eulerian orientation belonging to the subclass obtained by excluding alternating vertices. Bousquet-Mélou a…

Lecture hall position conjecture. If $\pi=\varphi(\lambda)$, then

Let $\mathfrak{S}_n$ be the set of permutations of $[n]$, and let $T_1,T_2,T_3$ be the statistics defined by … … … and let $S_{17}(p)$ be the largest $i$ such that $1,\dots,i$ appe…

Let $t\geq 1$, let $p$ be either $213$ or $321$, and call a permutation $p$-avoiding if it contains no subsequence order-isomorphic to $p$. Let $t$-stack-sortable permutations be t…

Let $\mathfrak{S}_n^t$ be the set of permutations of length $n$ that become the identity after $t$ applications of the stack-sorting operation, and let $\mathfrak{S}_n^t(p)$ denote…

Let $\alpha$ be a unit interval parking function, let $F$ denote the Foata transform, and let $\operatorname{spot}_\alpha$ denote the outcome permutation of $\alpha$. Foata transfo…

Let $t\geq 1$, $n\geq 0$, and $p\in\{213,321\}$. A permutation is $t$-stack-sortable if it can be sorted using $t$ applications of the stack-sorting operator, and it avoids a patte…

Let $\mathcal{T}_n(132)$ and $\mathcal{T}_n(321)$ denote the sets of shallow permutations of size $n$ avoiding the patterns $132$ and $321$, respectively. For a permutation $\pi$,…

Let $S_{n,d}$ be the complete split graph, and let $\mathsf{SortedRec}(S_{n,d})$ be the set of sorted recurrent configurations on $S_{n,d}$. For a configuration $c$, write…

Inverse-permutation conjecture. The map $g_{1,n}\circ g_{2,n}^{-1}$ maps every permutation to its inverse.

Fixed-point conjecture. There is only $1$ fixed point of $f_{4,n}\circ f_{1,n}^{-1}:S_n\to S_n$, namely the identity permutation, corresponding to the word $1...1$.

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…

Three symmetry conjectures for di-sk trees. Over di-sk trees, the following symmetries should hold:

A double-tandem walk is a walk for which the starting point is $(x_{start},y_{start})$, and $x_{min}$ and $y_{min}$ denote the minimal $x$- and $y$-coordinates visited during the w…

Tenner's conjecture. Iteration of the map $\phi_s\phi_t$ suffices to produce a bijection: for every $s$-regular, $t$-distinct partition $\lambda$, there exists an integer $\ell$ de…

Unified bijection conjecture. A unified bijection for essentially irreducible toroidal $d$-angulations should exist.

Let $Q$ be a standard Young tableau and let $V$ be its corresponding vacillating tableau. Let evacuation, also called the Schützenberger involution, act on $Q$, and let reversal ac…

Let standard Young tableaux be concatenated when their row lengths all have the same parity, and let vacillating tableaux of shape $\emptyset$ be concatenated in the corresponding…

Garsia–Milne iteration conjecture. In the worst case, $\Psi_n$ requires $\exp(n^{\Omega(1)})$ iterations.

Let $k$ and $n$ be positive integers, and let $\lambda=\lambda_1\lambda_2\ldots\lambda_\ell$ be a type with $\lvert\lambda\rvert$ denoting its size. A small $(k,k)$-Fuss-Schröder p…

Bijection conjecture. The following families are in bijection: hesitating excursions of length $2n$ in the quadrant; hesitating axis-walks of length $2n$ in the octant; and hesitat…

Let $\mathcal{F}$ be the set of all fillings of Young diagrams with positive integers. For a filling $\sigma$, let $\operatorname{inv}(\sigma)$ and $\operatorname{maj}(\sigma)$ den…

Doubled-chamber conjecture. The set of oscillating lattice walks of length $n$ in $W_k$ starting at $\delta$ and ending at the boundary