GINI INDEX ON GENERALIZED r-PARTITIONS

The Gini index of a set partition p of size n is defined as 1 - delta(pi)/n(2), where delta(pi) is the sum of the squares of the block cardinalities of pi. In this paper, we study the distribution of the delta 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 delta 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 d 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. (C) 2022 Mathematical Institute Slovak Academy of Sciences.