Generalized Fishburn numbers and torus knots

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...

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Bibliographic Details
Published in:arXiv.org 2020-10
Main Authors: Bijaoui, Colin, Boden, Hans U, Myers, Beckham, Osburn, Robert, Rushworth, William, Tronsgard, Aaron, Zhou, Shaoyang
Format: Article
Language:English
Subjects:
Congruences
Knots
Thermal expansion
Toruses
Online Access:Get full text
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Summary: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 \(\mathscr{F}_{t}(q)\).
ISSN:2331-8422
DOI:10.48550/arxiv.2002.00635