12 problems

Enumeration conjecture.

For each $n\geq 1$, let $a_n^{\{1,3\}}(\sigma)$ count length-$n$ permutations avoiding the pattern $\sigma$ whose cycle lengths belong to $\{1,3\}$, and let…

Expansion conjecture.

Let $[n]=\{1,\ldots,n\}$, let $S_n$ be a uniform $231$-avoiding permutation of $[n]$, and define its normalized graph by … View $\operatorname{gr}(S_n)$ as a random compact subset…

Let $\pi$ be a $123$-avoiding permutation, and let its boundary grid be the associated skew Young diagram. The downcore graph has the boxes of the boundary grid as vertices, with a…

Let $J(\ell,k)$ denote the number of $132$-avoiding permutations of length $\ell$ whose staircase encoding has $k$ nonzero boxes, and let $C_i$ denote the $i^{\text{th}}$ Catalan n…

Let $S_{n,k}$ be the quantity defined earlier in the paper, let $\rho$ and $\tau$ be patterns of length $4$, and let $d53e_n(\rho,\tau)$ denote the permutations in $d53e_n$ avoidin…

Let $m\geq4$ and let $\tau=134\ldots(m-1)2m$ be the consecutive pattern of length $m$. Let $T(x)$ be the cluster generating function specified by the preceding conjectural hypergeo…

Rightmost-entry conjecture. The rightmost numbers in rows numbered $2+3i$, for $i=0,1,\ldots$, are the Catalan numbers, and the rightmost numbers in rows numbered $1+3i$, for…

Let $S_n(q)$ denote the number of permutations of length $n$ that avoid a pattern $q$. For $m\geq 2$, define $M_{2m}$ to be the pattern $132\cdots(2m-1)(2m-2)2m$, and define…

Let $S_n(q)$ denote the number of permutations of length $n$ that avoid a pattern $q$. A pattern is layered if it consists of decreasing subsequences whose entries increase from on…

Narayana distribution conjecture. The path lengths have the Narayana distribution: