## Abstract

In this paper, we enumerate classes of partitions of [n] = {1, …, n} 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 {x} 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.

Original language | English |
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Pages (from-to) | 100-107 |

Number of pages | 8 |

Journal | Discrete Mathematics Letters |

Volume | 13 |

DOIs | |

State | Published - 2024 |

### Bibliographical note

Publisher Copyright:© 2024 the authors.

## Keywords

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

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics