Set partitions with colored singleton blocks

Toufik Mansour; Augustine O. Munagi; Mark Shattuck

doi:10.47443/dml.2024.009

Author(s)

Toufik Mansour Department of Mathematics, University of Haifa,3103301 Haifa,Israel

Augustine O. Munagi School of Mathematics, University of Witwatersrand,2050 Wits, Johannesburg,South Africa

Mark Shattuck Department of Mathematics, University of Tennessee,37996 Knoxville, Tennessee,USA

Abstract

In this paper, we enumerate classes of partitions of [n] = f1; : : : ; ng in which the singleton blocks are colored using a variable or fixed number of colors. We consider, more generally, the distribution of the statistic recording the number of colored singletons on r-partitions of [r + n] in which only singletons from [r + 1; r + n] may be colored. Among our results, it is shown by algebraic and bijective arguments that the number of partitions of [n] in which a singleton block fxg can come in one of x colors for each x is given by the n-th row sum of Lah numbers, yielding a new combinatorial interpretation for this sequence. Also, we show that the partitions of [n] in which each singleton is assigned one of s + 1 colors where s is fixed are equinumerous with the set of s-partitions of [s + n]. Generalizations in terms of r-partitions of both of these results and others are demonstrated.

Publication Details

Page(s) 100-107

DOI 10.47443/dml.2024.009

Published Journal: Discrete Mathematics Letters, Volume: 13, Issue: 0, Year: 2024

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Keywords

finite set partition, Lah distribution, singletons statistic, exponential generating function

References

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