222 problems

Pattern-avoidance conjecture. The map $\mathrm{hat}_{\max}$ restricts to the following bijections:

Transposed Catalan triangle formula. The number of such permutations is

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 $F_s(q)$ 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:

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…

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

Riehl's conjecture. The number of basketball walks of length $n$ starting at the origin and ending at altitude $1$ that never touch or pass below the $x$-axis equals the number of…

Duncan–Steingrímsson's conjecture. On modified ascent sequences, these six patterns are all Wilf-equivalent, and the enumeration of modified ascent sequences avoiding any one of th…

Strong Wilf-equivalence conjecture. Two binary words $p$ and $q$ are strongly Wilf-equivalent if and only if they are trivially equivalent.

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…

Equidistribution conjecture. The statistics $\operatorname{pix}$ and $\operatorname{fix}$ are equidistributed over $\mathfrak{S}_{n}(\Pi)$ for all $n\geq0$ if and only if $\Pi$ is…

Grassmannian enumeration conjecture. For every $n\geq 1$,

The Type B non-decreasing parking functions are the elements of the submonoid of $\operatorname{NDPF}_{2N}$ generated by $\mu_i=\pi_i\pi_{2N-i}$ for $1\leq i\leq N$, with…

Elizalde–Noy maximality conjecture. The increasing pattern $\sigma$ is the maximal pattern, in the sense that

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…

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…