12 problems

Circuit-free permutation conjecture. The permutation $w$ is circuit-free, and thus achieves the lower bound in Theorem, if and only if at least one of the following conditions is s…

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.…

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 $S_n$ 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…

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

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…

Let $R(\infty)$ be the set of all reduced words of permutations. Two reduced words are dual $EG$-equivalent when their recording tableaux under Edelman–Greene insertion agree, that…

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