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## Bulletin of the Polish Academy of Sciences. Mathematics

2015 | 63 | 2 | 123-140
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### Some Parity Statistics in Integer Partitions

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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 $λ₁ ≥ ⋯ ≥ λ_k$ may be enumerated according to descents $λ_i > λ_{i+1}$ while tracking the individual parities of $λ_i$ and $λ_{i+1}$. 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.
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123-140
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2015
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autor
• School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
autor
• Department of Mathematics, University of Haifa, 3498838 Haifa, Israel
autor
• School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
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