19 problems

Let ${\cal P}_n^k$ denote the set of partitions of $[n]$ into $k$ blocks, and let $\mathop{\rm mak}$, $\mathop{\rm lmak}$, $\mathop{\rm mak}{}'$ and ${\mathop{\rm lmak}}'$ be the f…

For a positive integer $m$, let $[m]=\{1,\dots,m\}$, and let $\sigma$ and $\tau$ be set partitions of $[m]$, with $\tau\ge\sigma$ meaning that every block of $\sigma$ is contained…

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…

Let $a(n)$ be the integer sequence from Theorem, and define its binary reduction by … A binary sequence is normal if every finite binary string of length $\ell\geq 1$ occurs in it…

A $(k,\ell)$-partition is a set partition of $\{1,2,\dots,k\ell\}$ with exactly $\ell$ blocks, each of size $k$. Let $S_{i,j}$ denote the canonical partially 2-intersecting family…

Let $\Gamma$ be a finite Abelian group of order $n$ with more than one involution. Let … be a partition of $n-1$ with $r_i\geq 3$ for $1\leq i\leq t$, where $t$ is any positive int…

Let $pi1 otinPik$ and $pi2 otinPil$ with $k otin l$. For a pattern $pi$, write $pi^c$ for its complement, and let $Pil(pi1)$ denote the set of length-$l$ patterns avoided jointly w…

An inversion sequence of length $n$ is a sequence $x=x_1x_2\cdots x_n$ satisfying $0\leq x_i<i$ for all $1\leq i\leq n$. A set partition of $[n]$ is enhanced 3-nonnesting (respecti…

For $k\geq2$, let $C_k(n)$ be the number of partitions of $[n]$ avoiding classical $k$-crossings, and let $E_k(n)$ be the number avoiding enhanced $k$-crossings. Binomial-transform…

For each $n\geq1$, let $\operatorname{{\bf I}}_n$ be the set of inversion sequences … Let $\operatorname{{\bf I}}_n(\geq,\geq,-)$ be the subset of inversion sequences for which the…

Chen–Lev conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exists an integer $l\ge1$ such that

Partition-matching exponent conjecture.

Strict ordering conjecture. If $k\geq 4$, $\tau\in\Pi_k$, and $\tau\neq\beta_k$, then

Let $[n]=\{1,2,\ldots,n\}$, and let $P$ be a set partition of $[n]$. Its front representation is the linear representation obtained by joining consecutive elements in each block. F…

Let $X_n$ and $Y_n$ be the random variables defined in the paper, representing the relevant crossing statistics on set partitions. Asymptotic Gaussianity conjecture. The distributi…

Let $C_n(r)$ denote the number of colored set partitions of $[n]$ with colors from a set of size $r$. A formal power series is D-finite if it satisfies a linear differential equati…

Let $A_{210}(n)$ be the number of ascent sequences of length $n$ avoiding $210$. A set partition of $\{1,2,\ldots,n\}$ is 3-noncrossing if it has no 3-crossing. The 210 counting co…

Let $k$-noncrossing set partitions be set partitions whose arc diagrams contain no $k$-crossing, and let their ordinary generating function count them by size. Bousquet-Mélou–Xin c…

Let $\Pi_n$ be the set of set partitions of $[n]$, and let $\operatorname{FCR}(\pi)$ and $\operatorname{FNE}(\pi)$ denote respectively the front crossing and front nesting statisti…