Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.3

Minimal r-Complete Partitions

Øystein J. Rödseth
Department of Mathematics
University of Bergen
Johs. Brunsgt. 12
N-5008 Bergen


A minimal r-complete partition of an integer m is a partition of m with as few parts as possible, such that all the numbers 1,..., rm can be written as a sum of parts taken from the partition, each part being used at most r times. This is a generalization of M-partitions (minimal 1-complete partitions). The number of M-partitions of m was recently connected to the binary partition function and two related arithmetic functions. In this paper we study the case r ≥ 2, and connect the number of minimal r-complete partitions to the (r+1)-ary partition function and a related arithmetic function.

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(Concerned with sequences A000123 A002033 A005704 A005705 A005706 A018819 A100529 A117115 and A117117 .)

Received May 14 2007; revised version received July 30 2007. Published in Journal of Integer Sequences, August 3 2007.

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