5 problems

Let symmetrized shuffling operators be the operators obtained by composing a move-and-randomize shuffling operator with its transpose. Reiner, Saliola and Welker also described an…

Let $\{\nu_k\}_{k\in\mathbb{N}}$ be the symmetrized shuffling operators $\nu_k=\pi_k^\top\pi_k$, where $\pi_k$ removes $k$ objects randomly and puts them on top in a random order.…

Let $\lambda$ and $\gamma$ be distinct partitions of $n$, both different from $(1^n)$ and $(n)$. For a partition $\lambda$, let $\nu_\lambda$ be the operator associated with the…

Let $W=\mathfrak{S}_n$ and let $\nu_{(k,1^{n-k})}$ be the matrix whose $(u,v)$-entry is the $k$-noninversion number of $v^{-1}u$, for $k=1,2,\ldots,n$. Integrality conjecture. The…

Let $W=\mathfrak{S}_n$ and let $nu_{\mathcal{O}}$ denote the operator associated with a conjugacy class of parabolic subgroups. The two known commuting families are…