23 problems

Dyer–Hohlweg conjecture. Each equivalence class under $\sim_m$ contains a unique minimal length element, and the set of these minimal length elements is the set of $m$-low elements…

Little's conjecture. The two tableaux agree:

For a positive integer $n$ and an affine permutation $w\in \widetilde{\mathfrak{S}}_n$, consider the higher-Bruhat-type theorem stated immediately before this conjecture, including…

For an affine permutation $w\in \widetilde{\mathfrak{S}}_n$, let $\mathcal{G}(w)$ be the directed graph whose vertices are commutation classes of reduced words for $w$, with edges…

Let $n\geq 1$ and let $\sigma\in\mathfrak{S}_{n+1}$. Let $R_{\bullet}(\sigma)$ denote the set of one-element commutation classes of reduced words of $\sigma$. The possible-cardinal…

Let $n\geq 1$ and let $\sigma\in\mathfrak{S}_{n+1}$. Write $R(\sigma)$ for the set of reduced words of $\sigma$ and $C(\sigma)$ for the set of commutation classes of reduced words.…

Fix $p\in(0,1)$, and for each $n\geq 2$ let $\mathsf{w}^{(n)}=(s_{n-1})(s_{n-2}s_{n-1})\cdots(s_1s_2\cdots s_{n-1})$ be the staircase reduced word, and let $v^{(n)}$ be the random…

Linear-ceiling formula conjecture. There exist real numbers $a_k$ and $b_k$ such that

Let $d53e$ be the symmetric group on $n$ letters, and let $w_0^{(k,i)}$ denote the family of permutations specified in the paper. For $d70e i d53e$, let $G(d70e)$ be the graph of r…

The nonnegative excess conjecture. For any $d70e i d53e$,

Let $d53e$ be the symmetric group on $n$ letters. For $d70e i d53e$, let $d4a1(d70e)$ be the set of reduced words of $d70e$ and let $d49e(d70e)$ be the set of braid classes of redu…

Let $W(B_n)$ be the Weyl group of type $B_n$, let $w\in W(B_n)$, let $G(w)$ be the graph of reduced words of $w$, and let $L_2(w)$ be the associated set of rank-two root subsystems…

Let $S_n$ be the symmetric group, let $w\in S_n$, let $G(w)$ be the graph of reduced words of $w$, let $L_2(w)$ denote the associated set of rank-two root subsystems, and let…

Let $\pi\in\mathfrak S_n$ be a permutation, and let $G_\pi$ and $L_2(\pi)$ be as above. 3412-pattern conjecture. If $\pi$ contains a pattern $3412$, then … The claim would give a n…

Let $\pi\in\mathfrak S_n$ be a permutation. Write $\iota_k=12\dots k$ and use $12[\cdot]$ and $21[\cdot]$ for permutation inflations. Lower-bound equality conjecture. The permutati…

For a permutation $\pi\in\mathfrak S_n$, let $L_2(\pi)$ denote the union of the sets of pairs and triples of inversions described above, and let $G_\pi$ be the graph associated wit…

Inversion-major-index equidistribution conjecture. For $n\geq 1$,

Stanley introduced symmetric functions indexed by permutations to enumerate reduced words. Edelman and Greene introduced balanced labelings of Rothe diagrams in connection with the…

Let $w_0=[n,n-1,\ldots,1]\in S_n$ be the longest permutation, let $p$ be its inversion number, and choose a reduced word $(a_1,\ldots,a_p)$ uniformly at random. Form the initial pr…

Stanley's conjecture. For arbitrary $\sigma\in S_n$, $|\operatorname{Red}(\sigma)|$ can be expressed as the number of standard Young tableaux of various shapes, possibly with multi…

Combinatorial Sylvester four-point conjecture. For every $n\geq 4$, the probability that $\mathsf{v}\in X$ is $1/4$.

Diameter bounds conjecture. For an element $w$ in the symmetric group $d54a_n$,

Let $R(w_0)$ denote the set of reduced expressions for the longest element $w_0$ in the symmetric group, and let $SYT(n^n)$ denote the set of standard Young tableaux of square shap…