The Gini index of a set partition π of size n is defined as 1 − δn(π2) , 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.
Bibliographical notePublisher Copyright:
© 2022 Mathematical Institute Slovak Academy of Sciences.
- combinatorial statistic
- Gini index
- Lah distribution
- set partition
ASJC Scopus subject areas
- Mathematics (all)