Some Parity Statistics in Integer Partitions

Aubrey Blecher ¹   ; Toufik Mansour ²   ; Augustine O. Munagi ¹

¹ School of Mathematics University of the Witwatersrand Private Bag 3 Wits 2050, Johannesburg, South Africa
² Department of Mathematics University of Haifa 3498838 Haifa, Israel

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) no. 2, pp. 123-140

Résumé

We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands $\lambda _1\ge \cdots \ge \lambda _k$ may be enumerated according to descents $\lambda _i>\lambda _{i+1}$ while tracking the individual parities of $\lambda _i$ and $\lambda _{i+1}$. There are two types of parity levels, $E=E$ and $O=O$, and four types of parity-descents, $E>E$, $E>O$, $O>E$ and $O>O$, where $E$ and $O$ represent arbitrary even and odd summands. We obtain functional equations and explicit generating functions for the number of partitions of $n$ according to the joint occurrence of the two levels. Then we obtain corresponding results for the joint occurrence of the four types of parity-descents. We also provide enumeration results for the total number of occurrences of each statistic in all partitions of $n$ together with asymptotic estimates for the average number of parity-levels in a random partition.

Détail

DOI : 10.4064/ba63-2-3

Keywords: study integer partitions respect classical word statistics levels descents subject prescribed parity conditions instance partition summands lambda cdots lambda may enumerated according descents lambda lambda while tracking individual parities lambda lambda there types parity levels types parity descents where represent arbitrary even odd summands obtain functional equations explicit generating functions number partitions according joint occurrence levels obtain corresponding results joint occurrence types parity descents provide enumeration results total number occurrences each statistic partitions together asymptotic estimates average number parity levels random partition