Armin Straub — International Journal of Number Theory — Volume 11, Number 5, 2015, Pages 1679-1690
Abstract
The Fishburn numbers \(\xi (n)\) are defined by the formal power series \[ \sum_{n \geq 0} \xi (n) q^n = \sum_{n \geq 0} \prod_{j = 1}^n (1 - (1 - q)^j). \] Recently, G. Andrews and J. Sellers discovered congruences of the form \(\xi (p m + j) \equiv 0\) modulo \(p\), valid for all \(m \geq 0\). These congruences have then been complemented and generalized to the case of \(r\)-Fishburn numbers by F. Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend to hold modulo prime powers.Download
| Link | Size | Description | Hits |
|---|---|---|---|
| fishburncongruences.pdf | 317.50 KB | Preprint (PDF, 13 pages) | 1964 |
| fishburnnumbers.m | 7.78 MB | Mathematica file containing the first 2500 Fishburn numbers | 2923 |
BibTeX
@article{fishburncongruences-2015,
author = {Armin Straub},
title = {Congruences for {F}ishburn numbers modulo prime powers},
journal = {International Journal of Number Theory},
year = {2015},
volume = {11},
number = {5},
pages = {1679--1690},
doi = {10.1142/S1793042115400175},
}