10 problems

Let $A_n(\sigma)$ be the set of permutations of $[n]$ that avoid the pattern $\sigma$ and are alternating. Lewis's conjecture. For $n\geq 0$ and…

Stanley's conjecture. For $n\geq4$ and $n\geq5$, respectively,

Callan's conjecture. The number of up-down permutations of even length fixed by reverse and complement equals the Springer numbers.

Let $\mathfrak{S}_n$ be the symmetric group, and define the two-sided alternating Eulerian polynomial by … Two-sided alternating gamma-expansion conjecture. For every $n\geq 1$, ……

Let $\mathfrak{S}_n$ be the symmetric group, and let $\operatorname{altdes}(\pi)$ and $\operatorname{altmaj}(\pi)$ denote the alternating descent number and alternating major index…

Let $\mathfrak{S}_n$ be the symmetric group, let $\operatorname{altmaj}(\pi)$ denote the alternating major index of $\pi\in\mathfrak{S}_n$, and let $m\in\mathbb{Z}$ satisfy…

Let $A_n(q)$ denote the set of alternating permutations of length $n$ avoiding the pattern $q$, and let $S_k$ be the symmetric group on $k$ letters. Alternating-avoidance minimizat…

Let $M(p)$ denote the permutation matrix of a pattern $p$, and set … Here $F_k\underset{1-\text{SASE}}{\sim}J_k$ denotes equivalence under $1$-SASE. SASE equivalence conjecture. Fo…

For a pattern $p$, let $A_m(p)$ be the set of alternating permutations of length $m$ that avoid $p$. Patterns $p$ and $q$ are equivalent for alternating permutations of a parity if…

Let $A_m(p)$ denote the set of alternating permutations of length $m$ that avoid the pattern $p$. Two patterns are equivalent for even-length alternating permutations when…