Set partitions and non-crossing partitions with ℓ-neighbors and ℓ-isolated elements

In this paper we introduce the notion of an ℓ-neighbor element of set partitions, that is, an element a in a block that contains ℓ +1 consecutive elements, among which is a. Elements that are not ℓ-neighbors are called ℓ-isolated elements. We explore combinatorial results to study these new statistics over the set partitions. In particular, we use combinatorial arguments, recurrence relations, and generating functions to describe our results. We also discuss possible relations with ribonucleic acids (RNA) structures.