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 $n\ge 0,$ $$\xi (pn+j)\equiv 0(\mathrm{mod}p).$$

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