Set partitions with colored singleton blocks

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.