68 problems

Let $W$ be a finite Coxeter group, let $C$ be a conjugacy class of involutions in $W$, and let $V_C$ be the real vector space with basis $\{a_w\mid w\in C\}$ equipped with the $W$-…

For each positive integer $n$, let $b_{n,k}$ denote the number of involutions in ${\mathfrak{S}}_n$ whose longest decreasing subsequence has length $k$, for $1\leq k\leq n$. A fini…

Symmetry conjecture. For all $m \ge 1$ and all $l_1, \ldots, l_m \ge 1$, the generating functions for the even involutions in $I(3412, [l_1, \ldots, l_m])$ and for the odd involuti…

Involution word tableau enumeration conjecture. If $n\geq 3$, then

A ring is called $*$-regular if it is regular and its involution is proper. It is directly finite if $ab=1$ implies $ba=1$, and unit-regular if for every element $a$ there is a uni…

Let $k\geq 1$ and let $\tau$ be a pattern. For patterns, write $\pi\stackrel{I}{\sim}\sigma$ when they are equally restrictive for involutions under pattern avoidance. Jaggard's co…

Nonnegative decomposition conjecture. For $n\geq 1$ and $k\geq 0$, the coefficients $a_{n,k}$ are nonnegative integers.

Area filling conjecture. There is a point $p\in S$ with

The conjecture. For all $n\geq 4$:

For a pattern $\tau$, let $\mathcal{I}_{n}(\tau)$ denote the number of involutions of length $n$ avoiding $\tau$, and write $\sigma\sim_{\mathcal{I}}\tau$ when the two patterns hav…

Let $\mu$ be a self-conjugate shape, and let $\mu_{sym}(\sigma)$ be the number of symmetric full placements on $\mu$ that avoid the pattern $\sigma$. Symmetric-placement prefix-exc…

Let $\sigma$ and $\tau$ be patterns, and write $\sigma\sim_{\mathcal{I}}\tau$ when they are equinumerous for avoidance by involutions, meaning that…

Let $\mathcal{I}_{n}(\tau)$ denote the number of involutions of length $n$ avoiding the pattern $\tau$, and let $M_n$ be the $n$th Motzkin number. Guibert's conjecture. For every…

Hart–Rowley ancestor property conjecture. Let $(W,S)$ be a finite Coxeter system, with $|W|<\infty$. If $w\in W$ and $w\neq 1$, then $w$ has the ancestor property.

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$. A stable odd-cycle two-line trade is an odd-cycle trade whose canonical selection remains compatible with reappli…

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$, let $\operatorname{sgn}(L)$ be the standard Alon–Tarsi sign, and let $\operatorname{Res}_n$ be a residual set. Re…

Let $G$ be an infinite group and let $S$ be a finite generating set. Define $\iota_S(G)$ as the involution probability associated with the word-ball exhaustion. Let…

Let $G$ be an infinite group equipped with a finitely additive measure $\mu$, and let $\iota_\mu(G)=\mu(\{x\in G\mid x^2=1\})$ be its involution probability. For a random finite su…

Let the algorithm in the paper be restricted to standard Young tableaux. Equivalence conjecture. The restricted algorithm is equivalent to the Sagan–Lee involution on standard Youn…

Let $n\geq 3$ be an odd integer, and set $k:=2n-1$. Let $\operatorname{Good}\Lambda_{4n,k}$ denote the set of good involutions associated with the twisted conjugation subquandle in…

Let $K$ be a knot, let $D$ be a diagram of $K$, and let … be the flip map, where $\eta$ is induced by the reflection of the diagram and $R$ is induced by the corresponding sequence…

A quandle is a set equipped with a binary operation satisfying the quandle axioms; it is connected when its inner automorphism group acts transitively, involutory when each right t…

Scalarity conjecture. Based on the authors' investigations, one has

Smith–Thom conjecture. The inequality

Bound on involution length. For every $w\in W$,