7 problems

Let $\Omega_n$ be the superspace ring with diagonal $\mathfrak{S}_n$-action, let $SR_n=\Omega_n/SI_n$ be its superspace coinvariant ring, and let $\mathbb{F}[\mathcal{OP}_n]$ be th…

Let $n\geq1$. For an ordered set partition $\Pi$ of $[n]$, let $|\Pi|$ be its number of blocks, let $\tau(\Pi)$ be its associated permutation, and let $w_\Pi(c)$ be the schedule nu…

Fix integers $r,n$ and $s$ with $1\le s\le rn-1$, and let … Consider decorated ordered set partitions $((L_1)_{l_1},\ldots,(L_k)_{l_k})$ with $1\in L_1$, satisfying … Let $w_P$ be…

Let $\Delta_r^n=\{x\in[0,r]^n:\sum x_i=r\}$ be the $(n-1)$-dimensional simplex, and call a decorated ordered set partition $((L_1)_{l_1},\ldots,(L_k)_{l_k})$ simplicial when…

Fix integers $1\le a<n$ and put $b=n-a$. Let $B_{a,b}=\{x\in[0,1]^n:\sum x_i=a\}$ be the hypersimplex, and let a decorated ordered set partition $((L_1)_{l_1},\ldots,(L_k)_{l_k})$…

Godbole et al.'s conjecture. For each pattern $p$ and each integer $k$, there exists a positive integer $n_0(k,p)$ such that for every $n\geq n_0(k,p)$,

Let $S_m$ be the symmetric group on $m$ letters, let $\rho\in S_m$ be a permutation, and let $\operatorname{op}_{n,k}(\rho)$ denote the number of ordered set partitions of $[n]$ in…