23 problems

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

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 $S_{n,d}$ 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…

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:

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…

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…

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…

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…

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…

For $k\geq 3$, let $IS_k$ be the set of irreducible $k$-shapes. For an irreducible $k$-shape $\lambda$, let $\overrightarrow{fr}(\lambda)=(t_1,\ldots,t_{k-2})\in\{0,1\}^{k-2}$ reco…

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…

Bijection conjecture. There is a bijection between beta$(0,1)$-trees on $n$ nodes fixed under $g$ and beta$(0,1)$-trees on $\lfloor n/2\rfloor+1$ nodes with one distinguished exces…

Gog–Magog bijection conjecture. There exists a bijection from Gog triangles of size $n$ to Magog triangles of size $n$, which maps $(n,k)$ Gog trapezoids to $(n,k)$ Magog trapezoid…

Let $\mathbb{P}^*$ be the set of finite words over the positive integers, let $\mathcal{F}(u)$ denote the set of words containing $u$ as a factor in generalized factor order, and l…

Replacement-bijection conjecture. If $s$ and $t$ are equivalent, then there is a sequence of top-down replacements, bottom-up replacements, and left--right reflections that produce…

Let $w$ and $u$ be permutations, and let $\theta^{(k)}$ denote a bijection on permutations for each $k$. The bijections are required to satisfy the properties stated in Proposition…

Let a unimodal polygon have two staircase regions with identical sides, possibly differing widths, and let the two boxes delimiting the staircase region be identified as in the pro…