Séminaire Lotharingien de Combinatoire, 86B.21 (2022), 12 pp.
Koustav Banerjee,
Sreerupa Bhattacharjee,
Manosij Ghosh Dastidar,
Pankaj Jyoti Mahanta and
Manjil P. Saikia
Parity Biases in Partitions and Restricted Partitions
Abstract.
Let po(n) (resp. pe(n)) denote the number of partitions of
n with more odd parts (resp. even parts) than even parts (resp.
odd parts). Recently, Kim, Kim, and Lovejoy proved that
po(n)>pe(n) for all n>2 and conjectured that
do(n)>de(n) for all n>19, where do(n) (resp.
de(n)) denotes the number of partitions into distinct parts
having more odd parts (resp. even parts) than even parts (resp. odd
parts). In this paper we provide combinatorial proofs for both the
result and the conjecture of Kim, Kim and Lovejoy. In addition, we
discuss other results on partitions with restricted parts where our
methods also work.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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