9 problems

Polynomial power-compositions determinant conjecture. For any positive integers $n$ and $k$,

Let ${\mathbf a}=(a_1,a_2,\dots,a_k)$ be a composition of $n$ into positive parts, and say that it contains $123$ if it has a subsequence order-isomorphic to $123$. Herb Wilf's con…

Let $w$ be a composition, ordered by the subword order, and let its principal downset consist of all compositions contained in $w$. A composition is associated with a layered permu…

Power-avoidance conjecture. The number of permutations $\pi\in\mathcal{S}_n$ that avoid $(312,321:\varnothing:2143)$ equals the number of compositions…

Let $C_n(\tau,\tau')$ be the set of all compositions of $n$ that avoid the patterns $\tau$ and $\tau'$, and let $c_n(\tau,\tau')=|C_n(\tau,\tau')|$. Compositional equivalence conje…

Let $R(n,k,l)$ denote the number of runs of $k$ of length $l$ over all compositions of $n$, and let $a(1,m)$ be the corresponding two-toned tiling count, interpreted as zero when i…

Let $a_s(r,n)$ denote the number of two-toned tilings of a $1\times(n+r+s)$ grid with $r$ red squares, where the last $s$ tiles must be white. For all $s,r,n\geq 1$, the conjecture…

Let $a(k,n,q)$ denote the number of compositions of $k$ with $n$ parts whose largest part is $q$, and let $\binom{n,q}{k}$ denote the coefficient of $x^k$ in…

Let ${\rm Max}_k(n)$ and ${\rm Min}_k(n)$ denote, respectively, the numbers of all compositions of $n$ having more than $k$ copies of the maximal and minimal part. Jaklič, Vitrih a…