11 problems

Dominance conjecture. If $T\subseteq T\circ\beta$, then $T\circ\beta$ dominates $T$.

Let ${\operatorname{Sort}}_n$ denote the Coxeter sorting time for permutations of size $n$. Asymptotic sorting-time conjecture. … This conjecture seeks an asymptotically sharp esti…

Let $\omega$ be a permutation, and let $P\in\operatorname{Pipes}(\omega)$. If $P$ is non-reduced, replace the redundant crossings with bump tiles. Up-elbow bound conjecture. Pipe…

Let $S_n$ be the symmetric group, let $\omega\in S_n$, and let $p_1(x_1,\ldots,x_n)$ and $p_2(x_1,\ldots,x_n)$ be monomials with nonzero coefficient in the Grothendieck polynomial…

Brenti–Carnevale–Tenner's conjecture. Each odd diagram class is rank-symmetric in the Bruhat order.

Let $n$ be a positive integer, let $k$ be a fixed positive integer with $1\leq k\leq n$, and let $\mathfrak{S}_n$ denote the symmetric group on $n$ elements. For…

Let $w$ be a permutation and let $a_{w,\lambda}$ denote the corresponding Edelman–Greene coefficient for a partition $\lambda$, with $f^{\lambda}$ the number of standard Young tabl…

Let $\mathscr{F}_n(\sigma)$ denote the set of Fishburn permutations of length $n$ avoiding the pattern $\sigma$. Fishburn-Wilf equivalence conjecture. The three pattern-avoidance c…

Let $n$ and $k$ be nonnegative integers, and let $C_q$ and $D_q$ be the quantities defined in Theorem 8 for compositions $q$. Write $\operatorname{Comp}_{k+1}(n-k)$ for the set of…

Let $w$ be a dominant permutation, meaning a 132-avoiding permutation, whose Lehmer code $\lambda=(c_1(w),\ldots,c_n(w))$ is the rectangular staircase shape $\lambda=\delta_d(b^a)$…

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…