Congruences for the Fishburn Numbers

TL;DR

This paper investigates congruence properties of Fishburn numbers, identifying specific primes and residue classes where these numbers vanish modulo p, revealing new divisibility patterns.

Contribution

It establishes new congruences for Fishburn numbers modulo primes, characterizing the residue classes where these numbers are divisible by p.

Findings

For half of the primes p, there exists a non-empty set T(p) with specific divisibility properties.

Fishburn numbers satisfy certain congruences modulo primes in the set T(p).

The paper identifies the structure of these congruences and their implications.

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 $n\geq 0,$ $$ \xi(pn+j)\equiv 0 \pmod{p}. $$