Two by two squares in set partitions

A partition πof a set S is a collection B₁, B₂, ..., B_k of non-empty disjoint subsets, alled blocks, of S such that âi=1kBi=S. $\begin{array}{} \displaystyle \bigcup-{i=1}kBi=S. \end{array}$ We assume that B₁, B₂, ..., B_k are listed in canonical order; that is in increasing order of their minimal elements; so min B₁ < min B₂ < â» < min B_k. A partition into k blocks can be represented by a word π= π₁π₂â»π_n, where for 1 ≤ j ≤ n, π_j â&circ;&circ; [k] and âi=1n{πi}=[k], $\begin{array}{} \displaystyle \bigcup-{i=1}n \{\pi-i\}=[k], \end{array}$ and π_j indicates that j â&circ;&circ; B_{π j}. The canonical representations of all set partitions of [n] are precisely the words π= π₁π₂â»π_n such that π₁ = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].