On singletons and adjacencies of set partitions

Abstract

The number of singleton blocks in all partitions of a set $\{a_1,\ldots ,a_n\}$ is known to be equal to the number adjacencies, that is, pairs of consecutively numbered elements $(a_i,a_{i+1})$ in a block. We give a generalization of this relation by introducing the $d$-adjacency which is a pair of elements $(a_i,a_j)$ satisfying $j-i=d>0$. It is proved that the number of $d$-adjacencies in all partitions is independent of $d$. Then we show that the number of $d$-adjacencies in noncrossing partitions is a function of $d$ by means of an exact formula.