104 problems

Let $f_{CS}(x)$ be the generating function for the cyclic shift strategy of length $n$, and let $f_S(x)$ be the generating function for a deranged strategy $S$, meaning a strategy…

Denominator conjecture. For all $k$, the rational generating function $F_k(x)$ can be written as

For natural numbers $k,n$, define $P^a_b=b!/(b-a)!$ whenever the indices are defined. Let $k,m,n$ be positive integers satisfying $1<2k\leq n-m$ and $1<2k\leq m$. Permutation ident…

Let $n$ be a positive integer. A cyclic V-shaped permutation of $[n]$ is a permutation with a unique local minimum that is also a cycle, and let the minimum occur at position $k$.…

Let $m\in\mathbb{P}$, and let $c(\operatorname{\mathcal{W}}(S_{2m}))$ be the convex hull of the Wachs permutations in $\mathbb{R}^{2m}$. Simplicity conjecture. The polytope…

USPN conjecture. For any permutation $\pi$ of any length, $\pi$-freeness has a non-adaptive one-sided $\varepsilon$-test making

Let $Y$ be the Young diagram under consideration, let $S_Y$ denote its transversals, and let $S_Y(213)$ denote the transversals avoiding the pattern $(213)$. For a transversal…

Let $S_n$ be the symmetric group, and let a displacement pattern be a tuple $\alpha=(a_1,a_2,\dots,a_n)$ recording the displacements of a permutation, with the entries distinct. Th…

The conjectured formula for $\mathfrak{D}_w$. For special $w$,

Let $n>2$ be prime, and define a permutation $g$ of $\mathbb Z_n$ by … Order permutations lexicographically, and let $\Psi(n)$ be the minimum number of collinear triples in a permu…

Let $S_n$ denote the set of permutations of $[n]=\{1,2,\ldots,n\}$, and let $S_n(\tau)$ be the set of permutations avoiding a pattern $\tau\in S_\ell$. Stanley–Wilf conjecture. For…

Monotonicity conjecture. The sequence

Bebeacua's conjecture. For every positive integer $n$,

Let $n$ and $t$ be positive integers, let $cSigma_n$ be the collection of permutations of $[n]$, and let $cA_k$ be the family … A family $\calF\subseteqcSigma_n$ is $t$-intersectin…

Monotonic growth-constant conjecture. For fixed $m$, the limit $\alpha_m$ exists, and

Let $S_n$ be the symmetric group, and let $A_n^{(m)}$ be the class … Here, $m$ is a fixed integer with $m\geq 1$. Finite-state structure conjecture. The class $A_n^{(m)}$ admits a…

Enumeration conjecture.

Let $n\geq 1$ and let $\sigma\in\mathfrak{S}_{n+1}$. Let $R_{\bullet}(\sigma)$ denote the set of one-element commutation classes of reduced words of $\sigma$. The possible-cardinal…

For $n\geq 1$, define $B(n,k)$ and $C(n,k)$ by … and … Here a simsun permutation is a permutation with the stated simsun property, and a big return is the stated statistic on an al…

Let $I_n(q,t)$ and $\widetilde{I}_n(q,t)$ be the two $t$-analogues of the tree inversion enumerator, let $\mathrm{PF}(n)$ denote the set of parking functions of length $n$, and for…

For a permutation $w$, let $\rp(w)$ denote its set of reduced pipe dreams, and order it by generalized chute moves. Rubey's lattice conjecture. This poset is a lattice. Certain spe…

Archer et al.'s conjectures. For the indicated values of $n$, the following enumerations hold:

Chen's conjecture. For any fixed $n$, the sequence $\ell_{n,1},\ell_{n,2},\ldots,\ell_{n,n}$ is log-concave. Equivalently, the distribution of $\ell_n(\sigma)$ for a uniformly chos…

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ and $\text{SC}_{3\underline{21}}$ be the corresponding vincular-pattern-avoiding…

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ and $\text{SC}_{3\underline{21}}$ be the corresponding vincular-pattern-avoiding…