43 problems

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 $k$-noncrossing partitions be set partitions containing no crossing of size $k$, and let the sequence count such partitions as a function of the underlying set size. Bousquet-M…

Let $k$ and $ell$ be integers, and let $\mathcal{U}_{k,\ell}$ denote the family of partitions of $[k\ell]$ into $\ell$ blocks of size $k$. A family…

Steingrímsson's conjecture. The generating functions of these statistics on ordered set partitions are given by either side of

Let $k>3$. The generating function of $k$-noncrossing partitions is the formal power series that enumerates $k$-noncrossing partitions by their size. Bousquet-Mélou and Xin's conje…

Let $[n]=\{1,2,\ldots,n\}$, and consider arbitrary set partitions of $[n]$ under the reflection action and rotation classes under the corresponding reflection action. The preceding…

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…

Let $n>0$, and let a 3-good partition of $[n]=\{1,\dots,n\}$ mean a partition whose parts have at most three elements and whose part sums are powers of $3$. The 3-good partition mu…

Let $m>1$ and $n>0$. A partition of $[n]=\{1,\dots,n\}$ is $m$-good if every part has at most $m$ elements and the sum of the elements in each part is a power of $m$. The 3-good pa…

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…

Highest-$x$ coefficient conjecture. For generic genus $g>0$,

Genus-three Bell-number and generating-function conjecture. The genus-three numbers satisfy

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…

Let $\Pi_n$ be the set of set partitions of $[n]$, and let $\lambda_1$ denote the second largest eigenvalue of the Burnside process on $\Pi_n$. Burnside-process eigenvalue conjectu…

Let $X_{t,k,\ell}$ be the graph with vertex set $\mathcal{U}_{k,\ell}$, where two partitions $P$ and $Q$ are adjacent if and only if every pair of blocks $P_i\in P$ and $Q_j\in Q$…

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 each $n\geq1$, let $\operatorname{{\bf I}}_n$ be the set of inversion sequences … An inversion sequence $e=(e_1,e_2,\ldots,e_n)$ is an $\mathcal{AW}$-inversion sequence if, for…

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.

Let $N=2n+1$ be a positive integer, and let $d_1,\ldots,d_n\in(\mathbb{Z}/N)^*$. A partition of $\mathbb{Z}/N-\{0\}$ into pairs is a collection of $n$ disjoint two-element subsets…