252 problems

Mallows unimodality conjecture. If $\pi$ is any consecutive permutation pattern, then the function $p\mapsto\mathbb{E}_{n,p}(\pi)$ is unimodal.

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Growth conjecture. For $n \geq 12$,

Let $M$ be a cyclic partial multiplication matrix containing $\ell$ non-empty cells. A coil is a permutation represented by a cyclic sequence of points following the cycle of non-e…

For each $n\in\mathbb{N}$, let $W_n$ be the set of permutations of size $n$, let $\operatorname{Simp}_n$ be the set of blockwise simple permutations, and define … where $S_n$ is th…

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…

Let $S_n$ be the set of permutations of $\\{1,2,\ldots,n\\\\}$, and let two permutations be equivalent when one can be obtained from the other by replacing an occurrence of the pat…

Symmetry conjecture. For all integers $n$ and $k$ satisfying $1\leq k\leq n$, we have

Existence conjecture. For every positive integer $x$ whose length belongs to an admissible residue class modulo $144$, there exists a 3-inflatable permutation of length $x$.

For $k\le n$, let $\pi_{n,k}$ be the subpermutation of $\kappa_n$ induced by the values $n+1,n+3,\dots,n+2k-1,1,3n+1,2,3n+2,\dots,n,4n,n+2,n+4,\dots,n+2k$. In particular,…

Circuit-free permutation conjecture. The permutation $w$ is circuit-free, and thus achieves the lower bound in Theorem, if and only if at least one of the following conditions is s…

Bóna's real-rootedness conjecture. The descent polynomial $W_{n,t}(x)$ has only real zeros for any integer $1 \le t \le n-1$.

Polynomial-growth conjecture. If $\pi$ avoids some $t$-permutation, then

A pin word is a word encoding a permutation through the pin sequence construction; the set of permutations corresponding to pin words forms a permutation class. Rational generating…

Let $M$ be a finite matrix defining a grid class $\operatorname{Grid}(M)$, and let $G(M)$ be the bipartite graph whose vertices represent the rows and columns of $M$, with an edge…

Let $M$ be a finite matrix whose entries specify the monotone permutation classes placed in the cells of a grid, and let $\operatorname{Grid}(M)$ denote the resulting grid class. A…

Let $X$ be a finite proper subset of $\bigcup_{j\geq 2}\mathfrak{S}_j$, meaning that $X$ does not contain both an identity permutation $12\cdots j$ and a reverse identity permutati…

The Stanley-Wilf and subsequence-constant conjecture. Both limit superiors are actual limits, and

Let $X$ be a non-empty set of permutations, and let $s_n(X)=\left|\operatorname{Av}(X)\cap\mathcal{S}_n\right|$, where $\operatorname{Av}(X)$ is the class of permutations avoiding…

Shortest common superpattern conjecture. For any $l\ge 1$,

Layered extremal word conjecture. If $\Pi$ is a set of layered patterns, then

Explicit generating-function form conjecture. The displayed representations hold for every $r\geq1$. The paper presents this as a stronger conjecture motivated by explicit results;…

Let $q=1(l+1)l\dots 2$ with $l\ge3$, and let the frequency sequence for $q$ be the sequence of pattern frequencies indexed by $n$. A frequency sequence has internal zeros if it con…

Let $\tau,\sigma\in S_n$, and let $S_k(\tau)$ denote the set of permutations in $S_k$ avoiding $\tau$. Say that $\tau$ is asymptotically smaller than $\sigma$ if…