GINI INDEX ON GENERALIZED r-PARTITIONS

Toufik Mansour, Matthias Schork, Mark Shattuck, Stephan Wagner

Research output: Contribution to journalArticlepeer-review

Abstract

The Gini index of a set partition π of size n is defined as 1 − δn2) , 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 languageEnglish
Pages (from-to)1129-1144
Number of pages16
JournalMathematica Slovaca
Volume72
Issue number5
DOIs
StatePublished - 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)

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