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

Generalized Fishburn numbers and torus knots

Number Theory 2021-02-04 v2 Combinatorics Geometric Topology

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

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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 $\mathscr{F}_{t}(q)$ 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 $\mathscr{F}_{t}(q)$ .

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@article{arxiv.2002.00635,
  title  = {Generalized Fishburn numbers and torus knots},
  author = {Colin Bijaoui and Hans U. Boden and Beckham Myers and Robert Osburn and William Rushworth and Aaron Tronsgard and Shaoyang Zhou},
  journal= {arXiv preprint arXiv:2002.00635},
  year   = {2021}
}

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

Generalized Fishburn numbers and torus knots

Abstract

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