## Abstract

The Gini index of a set partition π of size n is defined as 1 − ^{δ}_{n}^{(π}_{2}^{)} , where δ(π) is the sum of the squares of the block cardinalities of π. In this paper, we study the distribution of the δ statistic on various kinds of set partitions in which the first r elements are required to lie in distinct blocks. In particular, we derive the generating function for the distribution of δ on a generalized class of r-partitions wherein contents-ordered blocks are allowed and elements meeting certain restrictions may be colored. As a consequence, we obtain simple explicit formulas for the average δ value, equivalently for the average Gini index, in all r-partitions, r-permutations and r-Lah distributions of a given size. Finally, combinatorial proofs can be found for these formulas in the case r = 0 corresponding to the Gini index on classical set partitions, permutations and Lah distributions.

Original language | English |
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Pages (from-to) | 1129-1144 |

Number of pages | 16 |

Journal | Mathematica Slovaca |

Volume | 72 |

Issue number | 5 |

DOIs | |

State | Published - 1 Oct 2022 |

### Bibliographical note

Publisher Copyright:© 2022 Mathematical Institute Slovak Academy of Sciences.

## Keywords

- combinatorial statistic
- Gini index
- Lah distribution
- set partition

## ASJC Scopus subject areas

- Mathematics (all)