5 problems

Let $\lambda$ be a partition with $n$ parts, and let $M=(m_{i,j})_{1\le i,j \le n}\in\mathcal{M}_\lambda$ be a minimal matrix of shape $\lambda$. A matrix is Hankel when its entrie…

Let $n\geq 1$ and $k\in \{0,1,\dots,n(n-1)/2\}$. The quantity $\operatorname{upf}_{n,k}^{\operatorname{inv}}$ denotes the number of unit interval parking functions of length $n$ wi…

Fix $p\in(0,1)$, and for each $n\geq 2$ let $\mathsf{w}^{(n)}=(s_{n-1})(s_{n-2}s_{n-1})\cdots(s_1s_2\cdots s_{n-1})$ be the staircase reduced word, and let $v^{(n)}$ be the random…

Let $p\in[0,1)$ and let $\boldsymbol{w}=\boldsymbol{w}(n)\in S_n$ be the random permutation in the non-reduced model. Write $\operatorname{inv}(\boldsymbol{w})$ for its number of i…

Inversion-enumeration conjecture.