Congruences and relations for r-Fishburn numbers

TL;DR

This paper extends congruence results for Fishburn numbers to a broader sequence, proving new congruences modulo 23 and 5, and confirming some conjectures related to these sequences.

Contribution

It introduces generalized congruences for a wider class of sequences related to Fishburn numbers, including new proofs of conjectured congruences.

Findings

Proved a new congruence mod 23 for Fishburn numbers.

Confirmed the conjectured mod 5 congruence for a related sequence.

Extended and proved unpublished conjectures of Garthwaite and Rhoades.

Abstract

Recently Andrews and Sellers proved some amazing congruences for the Fishburn numbers. We extend their results to a more general sequence of numbers. As a result we prove a new congruence mod 23 for the Fishburn numbers and prove their conjectured mod 5 congruence for a related sequence. We also extend and prove some unpublished conjectures of Garthwaite and Rhoades.