Definability of Combinatorial Functions and Their Linear Recurrence Relations

Abstract

We consider functions of natural numbers which allow a combinatorial interpretation as density functions (speed) of classes of relational structures, s uch as Fibonacci numbers, Bell numbers, Catalan numbers and the like. Many of these functions satisfy a linear recurrence relation over $\mathbb Z$ or ${\mathbb Z}_m$ and allow an interpretation as counting the number of relations satisfying a property expressible in Monadic Second Order Logic (MSOL). C. Blatter and E. Specker (1981) showed that if such a function $f$ counts the number of binary relations satisfying a property expressible in MSOL then $f$ satisfies for every $m \in \mathbb{N}$ a linear recurrence relation over $\mathbb{Z}_m$. In this paper we give a complete characterization in terms of definability in MSOL of the combinatorial functions which satisfy a linear recurrence relation over $\mathbb{Z}$, and discuss various extensions and limitations of the Specker-Blatter theorem.