23 problems

Let $C_n$ be the cactus group acting on the set of standard Young tableaux of a partition, and let $A_N$ denote the alternating group on $N$ tableaux, where $N$ is the number of st…

Let $c$ be a constant. A partition $\lambda$ of $n$ boxes is called generic if its Young diagram fits into a $c\sqrt n$ by $c\sqrt n$ grid. Write $\operatorname{SYT}(\lambda)$ for…

Involutions conjecture. There exist involutions $\alpha,\delta:\operatorname{SYT}(n^3)\to\operatorname{SYT}(n^3)$ with the following properties:

Sulanke's conjecture. On $\operatorname{SYT}(n^3)$, each of the statistics $\st_1$ and $\st_2$ has a $3$-Narayana distribution.

General skew-shape asymptotic normality conjecture. Asymptotic normality for general skew shapes, rather than only block diagonal skew shapes, holds if and only if

Log-concavity conjecture. The polynomials $\SYT(\lambda)^{\maj}(q)$ are almost always log-concave for partitions $\lambda\vdash n$ for large $n$.

Near-unimodality conjecture. The polynomials $\SYT(\lambda)^{\maj}(q)$ are nearly unimodal but not unimodal for partitions $\lambda$ or $\lambda'$ in the following cases: any parti…

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 $\lambda$ and $\mu$ be partitions with $\mu\subseteq\lambda$, and let $f_{\lambda\setminus\mu,i}(q)$ denote the major-index generating polynomial for standard Young tableaux of…

Conjectured three-rowed formulas. The two displayed identities hold.

Involution word tableau enumeration conjecture. If $n\geq 3$, then

Bergeron–Favreau–Krob conjecture. For each $k$, there are polynomials $p_m(t)$ of degree at most $\lfloor\frac{k}{2}\rfloor$ such that $Y_k(t)$ is a solution of a linear differenti…

Two-row tableau unimodality conjecture. For all $n$, $k$, and $i$, the polynomial $f_{(n-k,k),i}(q)$ is symmetric and unimodal, with central term $ni/2$.

Fix a positive integer $n$, and let $\Theta$ be the set of partitions of $n$ all of whose column lengths are even. Let $\ell_{n,k}^{\Theta}$ denote the corresponding quantity assoc…

Let $\lambda\vdash n$ with $n\geq 1$, and let $a_{\lambda,0}$ denote the number of standard Young tableaux of shape $\lambda$ whose major index is congruent to $0$ modulo $n$. Sund…

Let $X$ and $Z$ be LR-fillings of the same type, with the skew shape $\beta\setminus\gamma$ a rook strip and $Z<_{\sf dom}X$. The first and second algorithms produce LR-fillings…

Let $\lambda$ and $\mu$ be partitions with $\lambda\vdash n$, and suppose that $\mu$ is a hook. Write $f^{\lambda/\mu}$ for the number of standard Young tableaux of skew shape…

Let $d$ be a positive integer, and let the admissible unit steps be … where $\{\epsilon_1,\dots,\epsilon_{d+1}\}$ is the standard basis of $\mathbb{R}^{d+1}$. Consider paths from t…

Inner tableau translation conjecture. If $S\lessdot_{weak}T$ is a covering relation in $SYT_n^R$ and $R'$ is obtained from $R$ by applying a single dual Knuth relation, then

Property of inner tableau translation. If

Even-row lattice-path conjecture.

Let $A_1,\ldots,A_k$ be standard Young tableaux with $\operatorname{sh}(A_i)=\alpha_i\vdash a_i$ for $i\in[k]$. Let $\mathcal{T}_n(A_1,\ldots,A_k)$ be the set of $k$-tuples…