Partition Identities I: Sandwich Theorems and Logical 0–1 Laws

Abstract

The Sandwich Theorems proved in this paper give a new method to show that the partition function $a(n)$ of a partition identity $$ {\bf A}(x) \ :=\ \sum_{n=0}^\infty a(n)x^n\ =\ \prod_{n=1}^\infty (1-x^n)^{-p(n)} $$ satisfies the condition RT$_1$ $$ \lim_{n\rightarrow \infty}{a(n-1)\over a(n)} \ =\ 1\,. $$ This leads to numerous examples of naturally occuring classes of relational structures whose finite members enjoy a logical 0–1 law.