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Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial

Florian Luca;  Augustine O. Munagi — 2014

Colloquium Mathematicae

We note that every positive integer N has a representation as a sum of distinct members of the sequence d ( n ! ) n ≥ 1 , where d(m) is the number of divisors of m. When N is a member of a binary recurrence u = u ₙ n ≥ 1 satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

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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