Generalized Fishburn numbers and torus knots

Colin Bijaoui; Hans U. Boden; Beckham Myers; Robert Osburn; William Rushworth; Aaron Tronsgard; Shaoyang Zhou

doi:10.1016/j.jcta.2020.105355

paper

Generalized Fishburn numbers and torus knots

arXiv:2002.00635 · doi:10.1016/j.jcta.2020.105355

Abstract

Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper, we prove prime power congruences for generalized Fishburn numbers. These numbers are the coefficients in the $1 - q$ expansion of the Kontsevich-Zagier series $F_{t} (q)$ for the torus knots $T (3, 2^{t})$ , $t \geq 2$ . The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new "strange identity" for $F_{t} (q)$ .

13 pages, to appear in the Journal of Combinatorial Theory, Series A