## Abstract

A partition πof a set S is a collection B_{1}, B_{2}, ..., 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 B_{1}, B_{2}, ..., B_{k} are listed in canonical order; that is in increasing order of their minimal elements; so min B_{1} < min B_{2} < â» < min B_{k}. A partition into k blocks can be represented by a word π= π_{1}π_{2}â»π_{n}, where for 1 ≤ j ≤ n, π_{j} âˆˆ [k] and âi=1n{πi}=[k], $\begin{array}{} \displaystyle \bigcup-{i=1}n \{\pi-i\}=[k], \end{array}$ and π_{j} indicates that j âˆˆ B_{π} _{j}. The canonical representations of all set partitions of [n] are precisely the words π= π_{1}π_{2}â»π_{n} such that π_{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].

Original language | English |
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Pages (from-to) | 29-40 |

Number of pages | 12 |

Journal | Mathematica Slovaca |

Volume | 70 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2020 |

### Bibliographical note

Publisher Copyright:© 2020 Mathematical Institute Slovak Academy of Sciences 2020.

## Keywords

- Bell numbers
- Generating functions
- Restricted growth functions
- Set partitions

## ASJC Scopus subject areas

- General Mathematics