Some Parity Statistics in Integer Partitions

Aubrey Blecher, Toufik Mansour, and Augustine O. Munagi. "Some Parity Statistics in Integer Partitions." Bulletin of the Polish Academy of Sciences. Mathematics 63.2 (2015): 123-140. <http://eudml.org/doc/281291>.

@article{AubreyBlecher2015,
abstract = {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.},
author = {Aubrey Blecher, Toufik Mansour, Augustine O. Munagi},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {partition; descent; level; generating function; formula},
language = {eng},
number = {2},
pages = {123-140},
title = {Some Parity Statistics in Integer Partitions},
url = {http://eudml.org/doc/281291},
volume = {63},
year = {2015},
}

TY - JOUR
AU - Aubrey Blecher
AU - Toufik Mansour
AU - Augustine O. Munagi
TI - Some Parity Statistics in Integer Partitions
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2015
VL - 63
IS - 2
SP - 123
EP - 140
AB - 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.
LA - eng
KW - partition; descent; level; generating function; formula
UR - http://eudml.org/doc/281291
ER -