4 problems

Sokal's functional-digraph conjecture. The following assertions hold: (i) $[\psi_{n,k}]_{n,k}$ is totally positive; (ii) $(\bm{\psi}_n(y))_{n\geq0}$ is coefficientwise Hankel-total…

Let $\mathbb{F}$ be the semiring of functional digraphs, with multiplication given by the product of functional digraphs. A functional digraph $X\in\mathbb{F}$ is prime if…

Let $\Psi_n(x,y)$ be the generating polynomial for functional digraphs on $[n]$, with weight $x$ for each cyclic vertex and weight $y$ for each component. Define…

Let $\psi_{n,k}$ be the number of functional digraphs on $[n]$ with $k$ cyclic vertices, let $\Psi_n(y)=\sum_{k=0}^n\psi_{n,k}y^k$, and let $\Psi=(\psi_{n,k})_{n,k\geq 0}$. A matri…