46 problems

Herzog's conjecture. If $I_2(G)=I_2(H)$, then $|G|=|H|$.

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

Makar-Limanov's involutional conjecture. If $D$ is infinite dimensional over $k$ and finitely generated as a division $k$-algebra, then there exist two symmetric elements in $D$ wh…

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 $W$-…

Let $I_3=123$ and $J_3=321$. For a permutation pattern $\tau$, prefix exchange over alternating involutions means equality of the avoidance-class cardinalities after adjoining any…

Let $\mathcal{AI}_{m}(\rho)$ and $\mathcal{RAI}_{m}(\rho)$ denote, respectively, alternating and reverse alternating involutions of size $m$ avoiding the pattern $\rho$, and let…

Guo–Zeng conjecture. For every $n\in\mathbb{N}$, there exist coefficients $a_{n,i}\in\mathbb{N}$ such that

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 $\mathcal{L}_n$ 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…

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…

Smith–Thom conjecture. The inequality

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

The Ancestor Conjecture. Every finite Coxeter group has the ancestor property.

Let $H$ be a finite group generated by elements of order $2$; such a group is called $\{2\}$-generated. The involution-generation mixability conjecture. Every finite $\{2\}$-genera…

Let $I^{\mathsf{vex}}_\infty$ be the set of vexillary involutions, and let $\mathcal{B}_{\mathsf{inv}}(z)$, $\mathcal{B}_{\mathsf{inv}}^+(z)$, and…

Let $\mathbb{C}_n$ be the Clifford algebra with generators $i_1,\ldots,i_{n-1}$, and let $\mathbb{Z}[i_1,\ldots,i_{n-1}]$ be the indicated Clifford order. A maximal order is closed…

Let $I_n^D$ denote the involutions in the type-D signed permutation group, and let $G({\mathcal S}_n^B,X_1^D\cup X_2^D)$ be the Cayley graph with generating sets … … Consider the r…

Let $AI_n(\rho)$ denote the set of alternating involutions of length $n$ avoiding the pattern $\rho$. Let $\tau$ be any permutation of $\{4,\ldots,m\}$, where $m\geq 4$. The 123τ–3…

Let $AI_n(\rho)$ and $RAI_n(\rho)$ denote, respectively, the sets of alternating and reverse alternating involutions of length $n$ avoiding the pattern $\rho$, and let $M_n$ be the…

Let the reciprocal and twinning operations be the two involutions on tetrahedra defined in the surrounding discussion. Their compositions in either order are not themselves involut…