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Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Aubrey Blecher;  Charlotte Brennan;  Toufik Mansour — 2012

Open Mathematics

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the...

Some Parity Statistics in Integer Partitions

Aubrey Blecher;  Toufik Mansour;  Augustine O. Munagi — 2015

Bulletin of the Polish Academy of Sciences. Mathematics

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 λ ₁ ≥ ⋯ ≥ 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...

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