Journal of Integer Sequences, Vol. 23 (2020), Article 20.5.6

Unordered Factorizations with k Parts


Jacob Sprittulla
Germany

Abstract:

We derive new formulas for the number of unordered (distinct) factorizations with k parts of a positive integer n as sums over the partitions of k and an auxiliary function, the number of partitions of the prime exponents of n, where the parts have a specific number of colors. As a consequence, some new relations between partitions, Bell numbers, and Stirling numbers of the second kind are derived.

We also derive a recursive formula for the number of unordered factorizations with k different parts and a simple recursive formula for the number of partitions with k different parts.


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(Concerned with sequences A008731 A045778 A050322 A050323 A116608 A122179 A122180.)


Received July 18 2019; revised versions received August 25 2019; August 28 2019; April 4 2020. Published in Journal of Integer Sequences, June 5 2020.


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