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Gini index on generalized r-partitions

  • Toufik Mansour , Matthias Schork , Mark Shattuck EMAIL logo and Stephan Wagner
Published/Copyright: October 16, 2022
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 72 Issue 5

Abstract

The Gini index of a set partition π of size n is defined as 1−δ(π)n2, 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.

Keywords: Gini index; set partition; Lah distribution; combinatorial statistic
MSC 2010: Primary 05A15; 05A18; 91B99

  1. (Communicated by Gejza Wimmer)

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Received: 2021-05-02
Accepted: 2021-09-08
Published Online: 2022-10-16
Published in Print: 2022-10-26

© 2022 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular Papers
  2. Generalized hyperharmonic number sums with reciprocal binomial coefficients
  3. Gini index on generalized r-partitions
  4. Multiplicative functions of special type on Piatetski-Shapiro sequences
  5. Strengthenings of Young-type inequalities and the arithmetic geometric mean inequality
  6. Generalizations of the steffensen integral inequality for pseudo-integrals
  7. Subordination-implication problems concerning the nephroid starlikeness of analytic functions
  8. Boundedness and almost periodicity of solutions of linear differential systems
  9. On variational approaches for fractional differential equations
  10. Approximity of asymmetric metric spaces
  11. Approximation theorems for the new construction of Balázs operators and its applications
  12. On η-biharmonic hypersurfaces in pseudo-Riemannian space forms
  13. Chen’s first inequality for hemi-slant warped products in nearly trans-Sasakian manifolds
  14. Induced mappings on symmetric products of Hausdorff spaces
  15. The Teissier-G family of distributions: Properties and applications
  16. A new extension of the beta generator of distributions
  17. A new family of compound exponentiated logarithmic distributions with applications to lifetime data
  18. On two correlated linear models with common and different parameters
  19. On some applications of Duhamel operators
https://doi.org/10.1515/ms-2022-0077
Keywords for this article
Gini index; set partition; Lah distribution; combinatorial statistic

Articles in the same Issue

  1. Regular Papers
  2. Generalized hyperharmonic number sums with reciprocal binomial coefficients
  3. Gini index on generalized r-partitions
  4. Multiplicative functions of special type on Piatetski-Shapiro sequences
  5. Strengthenings of Young-type inequalities and the arithmetic geometric mean inequality
  6. Generalizations of the steffensen integral inequality for pseudo-integrals
  7. Subordination-implication problems concerning the nephroid starlikeness of analytic functions
  8. Boundedness and almost periodicity of solutions of linear differential systems
  9. On variational approaches for fractional differential equations
  10. Approximity of asymmetric metric spaces
  11. Approximation theorems for the new construction of Balázs operators and its applications
  12. On η-biharmonic hypersurfaces in pseudo-Riemannian space forms
  13. Chen’s first inequality for hemi-slant warped products in nearly trans-Sasakian manifolds
  14. Induced mappings on symmetric products of Hausdorff spaces
  15. The Teissier-G family of distributions: Properties and applications
  16. A new extension of the beta generator of distributions
  17. A new family of compound exponentiated logarithmic distributions with applications to lifetime data
  18. On two correlated linear models with common and different parameters
  19. On some applications of Duhamel operators
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