On Certain McKay Numbers of Symmetric Groups
Annemily G. Hoganson, Thomas Jaklitsch

TL;DR
This paper investigates partition functions related to McKay numbers of symmetric groups, deriving generating functions, explicit formulas for small primes, divisibility properties, and Ramanujan-type congruences.
Contribution
It provides explicit formulas, divisibility results, and Ramanujan-type congruences for partition functions associated with McKay numbers, extending understanding of their arithmetic properties.
Findings
Explicit generating functions for p_ell(a;n) in terms of p_ell(0;n)
p_ell(a;n) is zero for almost all n when ell=2,3
p_ell(a;n) is positive for large n when ell > 3
Abstract
For primes and nonnegative integers , we study the partition functions where denotes the product of hook lengths of a partition . These partition values arise as the McKay numbers in the representation theory of the symmetric group. We determine the generating functions for in terms of and specializations of specific D'Arcais polynomials. For and , we give an exact formula for the and prove that these values are zero for almost all . For larger primes , the are positive for sufficiently large . Despite this positivity, we prove that is almost always divisible by for any integer . Furthermore, with these results we prove several Ramanujan-type…
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Taxonomy
Topicsgraph theory and CDMA systems · Coding theory and cryptography · Finite Group Theory Research
