On (non-)realizibility of Stirling numbers

Abstract: We say that a sequence $(a_n)_{n\in\mathbb{N}_+}$ of non-negative integers is realizable if there exists a set $X$ and a mapping $T : X \to X$ such that $a_n$ is the number of fixed points of $T^n$. For each $k \in\mathbb{N}_+$ and $j \in \{1,2\}$ we define a sequence $S^{(j)}_k =(S^{(j)}(n+k -1,k))_{n\in\mathbb{N}_+}$ , where $S^{(j)}(n,k)$ is the Stirling number of the $j$-th kind (in case of $j = 1$ we consider unsigned Stirling numbers). The aim of the talk is to prove that $S^{(2)}_k$ is realizable if and only if $k \in \{1,2\}$, while for $k \geq 3$ the sequence $S^{(2)}_k$ is almost realizable with a failure $(k-1)!$, i. e. $(k-1)!S^{(2)}_k$ is realizable. Moreover, I will show that for each $k \in\mathbb{N}_+$ the sequence $S^{(1)}_k$ is not almost realizable, i. e. for any $r \in\mathbb{N}_+$ the sequence $rS^{(1)}_k$ is not realizable.

The talk is based on a joint work with Tom Ward (Newcastle, UK).

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)