Some Parity Statistics in Integer Partitions

Aubrey Blecher, Toufik Mansour, Augustine O. Munagi

Research output: Contribution to journalArticlepeer-review


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≥⋯≥λk may be enumerated according to descents λi>λi+1 while tracking the individual parities of λi and λi+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.
Original languageEnglish
Pages (from-to)123-140
Number of pages18
JournalBulletin of the Polish Academy of Sciences Mathematics
Issue number2
StatePublished - 2015


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