21 problems

Completeness conjecture. The equivalences found in the study, including the zero and non-zero equivalences, form a complete collection of equivalences.

Qualitative sharpness conjecture. For each of the ordinary and weak Bruhat orders, the upper bound on the number of comparable pairs is qualitatively close to the actual number of…

Let $x(\pi)$ be the X-ray of a permutation $\pi$, and let $\delta(x(\pi))$ denote the cardinality of its degeneracy class, namely the number of permutations having the same X-ray.…

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

Let $S_n$ be the symmetric group, let ${\mathbb Z}_n$ act as in the source, and let ${\operatorname{minv}}$ be the associated statistic on the quotient $S_n/{\mathbb Z}_n$. Unimoda…

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…

Strong $q$-log-concavity conjecture. The polynomial sequence $\left(b_k(S;q)\right)_{k=\mathbf{h}(m)-m}^{\mathbf{h}(m)}$ is strongly $q$-log concave.

Cooper's lexicographic-minimality conjecture. The permutation $g$ is the lexicographically least permutation satisfying that lower bound. The paper presents this as an open conject…

Let $q>2$ be prime, let $\mathbb{F}$ be the finite field with $q$ elements, and let $\Psi(q)$ denote the best lower bound for the number of collinear triples in a permutation of…

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 $x_1^{i_1}x_2^{i_2}\ldots x_n^{i_n}$ be a monomial with nonzero coefficient and non-maximal degree in the Grothendiec…

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…

Elizalde–Roichman conjecture. If $B$ is fine, then

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 $S_n$ be the symmetric group, let $\mathcal{I}(S_n)$ denote its involutions, and let $\Gamma(x)$ and $\operatorname{Cyc}(y)$ be the sets associated with $x$ and $y$ in the pape…

Let $k$ be the smallest integer with $k\leq n$ such that there exist $i_1<i_2<\ldots<i_k\in [n]$ for which, for every collection of values $f_1,\ldots,f_k\in[n]$, the set of functi…

Let $\omega\in S_n$ be a permutation. An $\omega$-chamber weakly separated collection is a collection of subsets of $\{1,2,\ldots,n\}$ in the sense introduced by Leclerc and Zelevi…