Some Parity Statistics in Integer Partitions

• Autorzy: Aubrey Blecher , Toufik Mansour , Augustine O. Munagi
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 05A15: Exact enumeration problems, generating functions
• 05A17: Partitions of integers
• 05A16: Asymptotic enumeration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: 63
• Numer: 2
• Strony: 123-140

Daty

wydano

2015

Twórcy

autor

Aubrey Blecher

• School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

autor

Toufik Mansour

• Department of Mathematics, University of Haifa, 3498838 Haifa, Israel

autor

Augustine O. Munagi

• School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

Identyfikatory

DOI

10.4064/ba63-2-3