GINI INDEX ON GENERALIZED r-PARTITIONS

被引：0

作者：

Mansour, Toufik ^{[1
]}

Schork, Matthias ^{[2
]}

Shattuck, Mark ^{[3
]}

Wagner, Stephan ^{[4
,5
]}

机构：

[1] Univ Haifa, Dept Math, IL-3498838 Haifa, Israel

[2] Haindell 99, D-65843 Sulzbach, Germany

[3] Univ Tennessee, Dept Math, Knoxville, TN 37996 USA

[4] Uppsala Univ, Dept Math, Box 480, S-75106 Uppsala, Sweden

[5] Stellenbosch Univ, Dept Math Sci, Private Bag X1, ZA-7602 Matieland, South Africa

来源：

MATHEMATICA SLOVACA | 2022年 / 72卷 / 05期

关键词：

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

D O I：

10.1515/ms-2022-0077

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页码：1129 / 1144

页数：16

共 26 条

[1] THE GINI INDEX OF RANDOM TREES WITH AN APPLICATION TO CATERPILLARS [J].