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Minimal ***r*-Complete Partitions

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Øystein J. Rödseth

Department of Mathematics

University of Bergen

Johs. Brunsgt. 12

N-5008 Bergen

Norway

**Abstract:**

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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