TY - JOUR

T1 - On the parity of the number of partitions with odd multiplicities

AU - Sellers, James A.

AU - Zanello, Fabrizio

N1 - Publisher Copyright:
© 2021 World Scientific Publishing Company.

PY - 2021

Y1 - 2021

N2 - Recently, Hirschhorn and the first author considered the parity of the function a(n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a(2m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a(n) for all n≢7(mod 8). We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case n 7(mod 8), where, interestingly, the behavior of a(n) modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, a(8m + 7) is odd precisely 50% of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open.

AB - Recently, Hirschhorn and the first author considered the parity of the function a(n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a(2m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a(n) for all n≢7(mod 8). We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case n 7(mod 8), where, interestingly, the behavior of a(n) modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, a(8m + 7) is odd precisely 50% of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open.

KW - Partition function

KW - binary integer representation

KW - density odd values

KW - eta-quotient

KW - odd multiplicity

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U2 - 10.1142/S1793042121500573

DO - 10.1142/S1793042121500573

M3 - Article

AN - SCOPUS:85101785770

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

ER -