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Some Parity Statistics in Integer Partitions

Volume 63 / 2015

Aubrey Blecher, Toufik Mansour, Augustine O. Munagi Bulletin Polish Acad. Sci. Math. 63 (2015), 123-140 MSC: 05A17, 05A15, 05A16. DOI: 10.4064/ba63-2-3

Abstract

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.

Authors

  • Aubrey BlecherSchool of Mathematics
    University of the Witwatersrand
    Private Bag 3
    Wits 2050, Johannesburg, South Africa
    e-mail
  • Toufik MansourDepartment of Mathematics
    University of Haifa
    3498838 Haifa, Israel
    e-mail
  • Augustine O. MunagiSchool of Mathematics
    University of the Witwatersrand
    Private Bag 3
    Wits 2050, Johannesburg, South Africa
    e-mail

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