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 may be enumerated according to descents while tracking the individual parities of and . 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-2-3,
author = {Aubrey Blecher and Toufik Mansour and Augustine O. Munagi},
title = {Some Parity Statistics in Integer Partitions},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {63},
year = {2015},
pages = {123-140},
zbl = {1329.05021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-2-3}
}
Aubrey Blecher; Toufik Mansour; Augustine O. Munagi. Some Parity Statistics in Integer Partitions. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) pp. 123-140. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-2-3/