Two by two squares in set partitions

Abstract

AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn 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].