222 problems

Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for linear extensions of the rectangular poset with parameters $s,t$ that avoid the permutation pattern $\sigma$,…

Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for pattern-avoiding linear extensions of the rectangular poset with parameters $s,t$. The 2143 two-column invers…

Let $F_s(q)$ be defined by … and, for $s\ge 2$, … Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for pattern-avoiding linear extensions of the rectangular poset…

Additional enumeration conjectures. The following identities are conjectured:

Fine-sequence conjecture. For every $n\geq 1$,

Asymptotic extremal covariance conjecture.

Strict inequality conjecture. The inequality in is strict for all $m>0$. This would imply that $\operatorname{Var}(X_\tau)$ has order $2m-1$ in $n$, so the standard deviation of…

Let $S_{n,k}$ be the triangle defined by … for $1\leq k\leq n-2$, with $S_{n,n}=S_{n,n-1}=S_{n,n-2}$ for $n\geq 3$ and $S_{1,1}=S_{2,1}=S_{2,2}=1$. For a pair of patterns…

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…

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 $X$ 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…

A pattern $q$ of length $k$ is a permutation pattern, and let $S_n(q)$ denote the number of permutations of length $n$ avoiding $q$. Define its Stanley–Wilf limit by … A pattern is…

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…

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…

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 $s_n((3,2,1),r)$ denote the number of permutations of length $n$ containing exactly $r$ occurrences of the pattern $(3,2,1)$, and let … For each $r$, let $P_r$ and $Q_r$ denote…

The 321 four-occurrence generating-function conjecture.

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…

For a permutation $\tau$, let $S_n(\tau)$ denote the set of permutations in $S_n$ avoiding $\tau$. Suppose $\tau,\sigma\in S_4$. Strong ordering conjecture. If … for some $k$, then…

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…

Recursive criterion for record-equivalence. The permutations $\pi$ and $\sigma$ are record-equivalent if and only if all of the following hold: