19 problems

Let $P_n$ denote the number of permutations of $n$ whose cycle lengths are prime numbers, and let the asymptotic result in Theorem 1 have an error term measuring the difference bet…

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 $f_n^k(\sigma)$ denote the number of permutations of length $n$ associated with a pattern $\sigma\in S_3$ and fixed by the $k$th iterate of the fundamental bijection. Tripling…

Let $f_n^k(\sigma)$ denote the number of permutations of length $n$ associated with a pattern $\sigma\in S_3$ and fixed by the $k$th iterate of the fundamental bijection. Power-of-…

Eventual constancy conjecture. For each $\sigma\in\{213,132\}$ and $k\geq1$, $f_n^k(\sigma)$ is eventually constant as $n$ becomes large. The paper conjectures that for $n\geq8$ th…

Let $t_n^k(\tau)$ denote the number of permutations of length $n$ avoiding the pattern $\tau$ that are fixed by the $k$th iterate of the fundamental bijection. For $ngeq2$, the con…

Let $C_n(231,1432: 231)$ be the set of permutations in $S_n$ that avoid the chain $(231,1432: 231)$, and let $c_n(231,1432: 231)$ denote its cardinality. Thus, the permutation avoi…

Let $f_{n,d}^{(2)}(123)$ denote the number of permutations of size $n$ that avoid the pattern $123$ and have stabilized interval parameter $d$. Write $n=2m+r$ with $r\in\{0,1\}$ an…

Let $A(n)$ denote the number of anti-coprime permutations of $[n]$, meaning permutations whose values are not coprime to their arguments. Let $cgamma$ denote the Euler–Mascheroni c…

Let $C(n)$ denote the number of permutations $csigma$ of $[n]$ such that $cgcd(j,csigma(j))=1$ for every $jcin[n]$. After defining constants $c_k$ for the corresponding restriction…

Let $F=(O,T_2,T_3,\dots,T_r)$ be a sequence of trees encoding a pinnacle set, where $O$ is the one-node tree, and let $p_n(F)$ denote the corresponding enumeration. Forest factoriz…

For a finite set $S$ of pinnacle values, let … Write $S=\{s_1,\dots,s_p\}$ with $n\geq s_p$, define … and set … Define abstract expressions by $E_1=x_{1,3}+x_{1,1}$,…

For $n\geq 1$, let $AV_n(23\text{-}1\text{-}4)$ be the permutations of length $n$ avoiding the generalized pattern $23\text{-}1\text{-}4$, and let $k$ be the number of right-to-lef…

For each $n$, let $AV_n(23\text{-}1\text{-}4)$ and $AV_n(1\text{-}{23}\text{-}4)$ denote the permutations of length $n$ avoiding the indicated generalized patterns. Shattuck's conj…

Zeilberger's conjecture. Not even God knows $s_{1000}(1 3 2 4,0)$.

For fixed integers $r\geq 0$ and $s\geq 0$, let $a(n)$ denote the number of permutations of length $n$ having exactly $r$ occurrences of the pattern $123$ and exactly $s$ occurrenc…

Let $B_n$ be the Boolean algebra of subsets of an $n$-element set, and let $B_n\setminus\{\emptyset\}$ be its truncation with the minimum element removed. Let $C_{n-1}$ be a chain…

Let $D$ and $E$ be the operators, and let $\langle W|$ and $|V\rangle$ be the vectors, occurring in the matrix formula for the refined enumeration; let $y$ and $q$ be indeterminate…