Definability of Combinatorial Functions and Their Linear Recurrence Relations

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.

On the Reconstruction of Graph Invariants

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U -polynomial, the universal edge elimination polynomial xi and the colore