Congruences for the Fishburn Numbers

Abstract

The Fishburn numbers, $\xi(n),$ are defined by a formal power series expansion $$ \sum_{n=0}^\infty \xi(n)q^n = 1 + \sum_{n=1}^\infty \prod_{j=1}^n (1-(1-q)^j). $$ For half of the primes $p$, there is a non--empty set of numbers $T(p)$ lying in $[0,p-1]$ such that if $j\in T(p),$ then for all $n\geq 0,$ $$ \xi(pn+j)\equiv 0 \pmod{p}. $$