Parity bias in partitions

Byungchan Kim; Eunmi Kim; Jeremy Lovejoy

doi:10.1016/j.ejc.2020.103159

Parity bias in partitions

Byungchan Kim, Eunmi Kim, Jeremy Lovejoy

School of Natural Sciences

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

Let p_o(n) denote the number of partitions of n with more odd parts than even parts and let p_e(n) denote the number of partitions of n with more even parts than odd parts. Using q-series transformations we find a generating function for p_o(n)−p_e(n), which implies that p_o(n)>p_e(n) for all positive integers n≠2. Using combinatorial mappings we prove a stronger result, namely that for all n>7 we have 2p_e(n)<p_o(n)<3p_e(n). Finally, using asymptotic methods we show that p_o(n)∕p_e(n)→1+2 as n→∞. We also examine related properties for two other types of partitions.

Original language	English
Article number	103159
Journal	European Journal of Combinatorics
Volume	89
DOIs	https://doi.org/10.1016/j.ejc.2020.103159
State	Published - Oct 2020

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abstract = "Let po(n) denote the number of partitions of n with more odd parts than even parts and let pe(n) denote the number of partitions of n with more even parts than odd parts. Using q-series transformations we find a generating function for po(n)−pe(n), which implies that po(n)>pe(n) for all positive integers n≠2. Using combinatorial mappings we prove a stronger result, namely that for all n>7 we have 2pe(n)o(n)<3pe(n). Finally, using asymptotic methods we show that po(n)∕pe(n)→1+2 as n→∞. We also examine related properties for two other types of partitions.",

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AU - Lovejoy, Jeremy

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AB - Let po(n) denote the number of partitions of n with more odd parts than even parts and let pe(n) denote the number of partitions of n with more even parts than odd parts. Using q-series transformations we find a generating function for po(n)−pe(n), which implies that po(n)>pe(n) for all positive integers n≠2. Using combinatorial mappings we prove a stronger result, namely that for all n>7 we have 2pe(n)o(n)<3pe(n). Finally, using asymptotic methods we show that po(n)∕pe(n)→1+2 as n→∞. We also examine related properties for two other types of partitions.

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JO - European Journal of Combinatorics

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Parity bias in partitions

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