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

100%
Luca F., Munagi A.
Colloquium Mathematicum
|
2014
|
tom 134
|
nr 2
193-209
EN
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 $u = {uₙ}_{n≥1}$ 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.
2
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Some Parity Statistics in Integer Partitions

81%
Blecher A., Mansour T., Munagi A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
tom 63
|
nr 2
123-140
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.
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