Two by two squares in set partitions

Margaret Archibald (1) , Aubrey Blecher (1) , Charlotte Brennan (1) , Arnold Knopfmacher (1) , Toufik Mansour (2)

1 The John Knopfmacher Centre for Applicable Analysis and Number Theory [Johannesburg]
2 Department of Mathematics [Haïfa]

A partition $\pi$ of a set $S$ is a collection $B_1, \,B_2,\,\ldots, \,B_k$ of non-empty disjoint subsets, called blocks, of $S$ such that $\bigcup _{i=1}^kB_i=S$. We assume that $B_1, \,B_2,\,\ldots, \,B_k$ are listed in increasing order of their minimal elements, that is, $\textrm{min}\,B_1<\textrm{min}\,B_2<\cdots < \textrm{min}\,B_k$. A partition into $k$ blocks can be represented by a word $\pi=\pi_1\pi_2\cdots\pi_n$, where for $1 \leq j \leq n, \pi_j \in [k]$ and $\bigcup _{i=1}^n \{\pi_i\}=[k]$, and $\pi_j$ indicates that $j \in B_{\pi_j}$. The canonical representations of all set partitions of $[n]$ are precisely the words $\pi=\pi_1\pi_2\cdots\pi_n$ such that $\pi_1=1$, 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 $[n]$.

Mots clés: Set partitions; restricted growth functions; generating functions; Bell numbers

Domaines: Combinatoire [math.CO]

Margaret Archibald, Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, Toufik Mansour. Two by two squares in set partitions. 2016. ⟨hal-01276495⟩