Partition identities. I: Sandwich theorems and logical 0-1 laws (Q1883665)

Summary: The Sandwich Theorems proved in this paper give a new method to show that the partition function \(a(n)\) of a partition identity \[ A(x):= \sum^\infty_{n=0}a(n)x^n=\prod^\infty_{n=1} (1-x^n)^{-p(n)} \] satisfies the condition \(\text{RT}_1\) \[ \lim_{n\to\infty} \frac {a(n-1)} {a(n)}=1. \] This leads to numerous examples of naturally occurring classes of relational structures whose finite members enjoy a logical 0-1 law.