7 problems

Buck et al.'s three-run enumeration conjecture. The number of flattened $2$-Stirling permutations of order $n$ with exactly three runs equals the displayed sum.

Let $\mathcal{Q}_n$ denote the set of Stirling permutations of order $n$. A flattened Stirling permutation is a permutation in the corresponding flattened set, and let…

Let $\mathcal{Q}_n$ denote the set of Stirling permutations of order $n$, and let $\operatorname{flat}_3(\mathcal{Q}_n)$ be the set of those with exactly three runs. Three-run enum…

Linear coefficient conjecture. For all $r\geq2$ and $k\geq1$, the number of permutations in $\mathcal{P}_{r,k}$ with exactly one peak—equivalently, the number of permutations in…

Let $r(x)=\sqrt{\frac{1+x}{1-x}}$ and define $\mathscr{R}_n(x)=\sum_{k=0}^{2n-1}\mathscr{R}(n,k)x^k$ by … The recurrence is … with $\mathscr{R}_0(x)=1$ and $\mathscr{R}_1(x)=x$. Hu…

Let $n\geq 1$ and $0\leq i\leq n$. Write … for the trivariate extension of $A_{n,i}(y)$. A trivariate polynomial $p(x,y,z)=\sum_i s_i(x,y)z^i$ is partial $\gamma$-positive when eve…

Let $T_n(x)$ be the polynomial defined by the exponential generating function … For polynomials with only real coefficients and only imaginary zeros, say that $f(x)$ separates…