Partitions weighted by the number of two types of parts
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We confirm the conjecture proposed by ourselves and J. Lovejoy that for all $n>9$ \[ p'_e(n) > p'_o(n) \] holds, where $p'_e(n)$ (respectively, $p'_o(n)$) is the number of partitions of $n$ having an even (respectively, odd) number of odd parts larger than twice of the number of even parts. Moreover, we examine the connections between the number of partitions weighted by the number of two types of parts and partition functions from the literature on the theory of partitions.
Abstract
We confirm the conjecture proposed by ourselves and J. Lovejoy that for all $n>9$ \[ p'_e(n) > p'_o(n) \] holds, where $p'_e(n)$ (respectively, $p'_o(n)$) is the number of partitions of $n$ having an even (respectively, odd) number of odd parts larger than twice of the number of even parts. Moreover, we examine the connections between the number of partitions weighted by the number of two types of parts and partition functions from the literature on the theory of partitions.
참고문헌 (8)
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