27 problems

Let $G$ be a finite simple transitive permutation group of degree $n$, and let $\delta(G)$ denote the proportion of elements of $G$ that are derangements. Boston–Shalev conjecture.…

Isbell's conjecture. If $a$ is sufficiently large compared to $b$, then $G$ has a derangement whose order is a power of $p$.

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

Let $d_k(n)$ and $d_k^*(n)$ denote the two permutation statistics defined in the paper, and let $D_n$ denote the number of derangements (permutations without fixed points) in the s…

Let ${\mathfrak{D}}_n$ be the set of derangements in the symmetric group ${\mathfrak{S}}_n$, and let ${\mathbf{R}}({\mathfrak{D}}_n)$ be the associated graded module. For…

Let $\mathcal{C}$ be a permutation class, and let $\mathcal{C}^{\circ}$ be the set of derangements in $\mathcal{C}$. Write $\mathcal{C}_n$ and $\mathcal{C}^{\circ}_n$ for their mem…

Let $C_n$ be the $n$th Catalan number, and let $d_n$ be the number of derangements among the $123$-avoiding permutations of length $n$. The source considers an asymptotic form…

For each $n$, let $G_n(t)=\sum_{\rho}t^{\operatorname{exc}(\rho)}$, where the sum is over $123$-avoiding derangements of length $n$ and $\operatorname{exc}(\rho)$ is the number of…

Derangement proportion conjecture. The following formulas hold:

Let $G$ be a finite group acting on a finite set $\Omega$, and let a derangement be an element of $G$ that fixes no point of $\Omega$. Suppose that $G$ has exactly two orbits on…

Let $G$ be a finite transitive permutation group of degree $n$, and let $D(G)$ be the subgroup of $G$ generated by all derangements. Bailey–Cameron–Giudici–Royle's conjecture. If…

Let $G$ be a primitive affine permutation group of degree $n$, and suppose that $G$ is non-Frobenius. Write $\delta(G)$ for the proportion of elements of $G$ that are derangements.…

Alternating-group derangement bound conjecture. The following hold:

Alternating-group derangement-class square conjecture. In this case,

Derangement-class product conjecture. There exist conjugacy classes $C$ and $D$ of derangements such that

Larsen--Shalev--Tiep conjecture. Every finite simple transitive group satisfies

Soluble-stabiliser derangement conjecture. The optimal constant for all transitive groups with soluble point stabilisers is

Equal-orbit derangement conjecture. If $G$ has two orbits of size exactly $n/2$, then $G$ contains a derangement.

Spiga's conjecture. If

Almost simple fixer-ratio conjecture. We have

Let $n>1$ be an odd integer, let $S_n$ be the symmetric group on $\\{1,\ldots,n\\}$, and let $D(n-1)$ be the set of permutations $\tau$ of $\\{1,\ldots,n-1\\}$ with $\tau(j)\ne j$…

Let $G$ be a nontrivial finite transitive $2$-closed permutation group. A derangement is a permutation that fixes no point, and a derangement has prime order when its order is a pr…

Let $G$ be a transitive finite permutation group, and let $D(G)$ be the subgroup generated by the derangements in $G$. The derangement quotient realization conjecture. Every finite…

Let $G$ be a primitive permutation group of degree $n$, and let $D(G)$ be the subgroup generated by the derangements in $G$. A Frobenius group is a transitive permutation group who…

Let $G$ be a transitive permutation group of degree $n$, and let $D(G)$ be the subgroup generated by the derangements in $G$. A derangement is a permutation fixing no point. The de…