33 problems

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…

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

Stanley's conjecture. For every $n\geq 0$,

Let the algorithm in the paper be restricted to standard Young tableaux. Equivalence conjecture. The restricted algorithm is equivalent to the Sagan–Lee involution on standard Youn…

Quasipolynomial divisibility conjecture. For every fixed tableau $T$, there exists a quasipolynomial in $n$ such that, for $n$ sufficiently large, the orbit length of $T[n]$ always…

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.

Local limit conjecture. For every partition $clambda cvdash n$, uniformly in $n$ and over all integers $k$,

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

Local limit conjecture. Uniformly for all $n$ and all integers $k$,

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 related-shape formulas. All three displayed identities hold.

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

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…

Let $\lambda/\mu$ be a sequence of skew Young diagrams with $|\lambda/\mu|=N$, whose diagrams $\lambda$ and $\mu$ have a limit shape $\psi/\phi$ under $1/\sqrt{N}$ scaling in both…

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 $ni/2$.

Asymptotic constant conjecture. There is a constant $c=c(\overline\pi)$ such that

Krattenthaler–Panova conjecture. The number of tableaux is

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…