Geodesic
Minimal \(r\)-complete partitions
Journal of integer sequences, Tome 10 (2007) no. 8
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,$\dots , rm$ 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 $r \geq 2$, and connect the number of minimal $r$-complete partitions to the $(r+1)$-ary partition function and a related arithmetic function.
Classification : 11P81, 05A17
Keywords: complete partitions, M-partitions, (r + 1)-ary partitions 
@article{JIS_2007__10_8_a7,
     author = {R{\o}dseth,  {\O}ystein J.},
     title = {Minimal \(r\)-complete partitions},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {8},
     zbl = {1145.05002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_8_a7/}
}
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Rødseth,  Øystein J. Minimal \(r\)-complete partitions. Journal of integer sequences, Tome 10 (2007) no. 8. http://geodesic.mathdoc.fr/item/JIS_2007__10_8_a7/