Some Parity Statistics in Integer Partitions

• Autorzy: Blecher, A. , Mansour, T. , Munagi, A. O.
• 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.

Słowa kluczowe

EN

partition; descent,; level; generating function; formula

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 2
• Strony: 123--140
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Blecher, A.

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

autor

Mansour, T.

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

autor

Munagi, A. O.

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

Bibliografia

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• [6] P. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J. 12 (2006), 439-454.
• [7] P. Grabner, A. Knopfmacher and S. Wagner, A general asymptotic scheme for the analysis of partition statistics, Combin. Probab. Comput. 23 (2014), 1057-1086.
• [8] P. Erdős and J. Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8 (1941), 335-345.
• [9] A. Knopfmacher and N. Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
• [10] A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math. 308 (2008), 2492-2501.
• [11] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, https://oeis.org/, 2006.

Identyfikatory

DOI

10.4064/ba63-2-3