Startseite Gini index on generalized r-partitions
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Gini index on generalized r-partitions

  • Toufik Mansour , Matthias Schork , Mark Shattuck EMAIL logo und Stephan Wagner
Veröffentlicht/Copyright: 16. Oktober 2022
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 72 Heft 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)

References

[1] BALAJI, H. — MAHMOUD, H.: The Gini index of random trees with an application to caterpillars, J. Appl. Probab. 54 (2017), 701–709.10.1017/jpr.2017.28Suche in Google Scholar

[2] BELBACHIR, H. — BELKHIR, A.: Cross recurrence relations for r-Lah numbers, Ars Combin. 110 (2013), 199–203.Suche in Google Scholar

[3] BRODER, A. Z.: The r-Stirling numbers, Discrete Math. 49 (1984), 241–259.10.1016/0012-365X(84)90161-4Suche in Google Scholar

[4] CHEON, G.-S. — JUNG, J.-H.: r-Whitney numbers of Dowling lattices, Discrete Math. 312 (2012), 2337–2348.10.1016/j.disc.2012.04.001Suche in Google Scholar

[5] DOMICOLO, D. — MAHMOUD, H.: Degree-based Gini index for graphs, Probab. Engrg. Inform. Sci. 34 (2020), 157–171.10.1017/S0269964819000044Suche in Google Scholar

[6] FARRIS, F. A.: The Gini index and measures of inequality, Amer. Math. Monthly 117 (2010), 851–864.10.4169/000298910x523344Suche in Google Scholar

[7] FLAJOLET, P. — SEDGEWICK, R.: Analytic Combinatorics, Cambridge University Press, 2009.10.1017/CBO9780511801655Suche in Google Scholar

[8] GINI, C.: Variabilitá e Mutabilitá: Contributo allo Studio delle Distribuzioni e delle Relazioni Statistiche, C. Cuppini, Bologna, 1912.Suche in Google Scholar

[9] GOSWAMY, S. — MURTY, C. A. — DAS, A. K.: Sparsity measure of a network graph: Gini index, Inform. Sci. 462 (2018), 16–39.10.1016/j.ins.2018.05.044Suche in Google Scholar

[10] GRAHAM, R. L. — KNUTH, D. E. — PATASHNIK, O.: Concrete Mathematics: A Foundation for Computer Science, second edition, Addison-Wesley, Boston, 1994.Suche in Google Scholar

[11] HSU, L. C.–SHIUE, P. J.-S.: A unified approach to generalized Stirling numbers, Adv. Appl. Math. 20 (1998), 366–384.10.1006/aama.1998.0586Suche in Google Scholar

[12] KOPITZKE, P.: The Gini index of an integer partition, J. Integer Seq. 23 (2020), Art. ID 20.9.7.Suche in Google Scholar

[13] MANSOUR, T.: Combinatorics of Set Partitions, Chapman   Hall/CRC, Taylor   Francis Group, Boca Raton, 2012.10.1201/b12691Suche in Google Scholar

[14] MANSOUR, T. — SCHORK, M. — SHATTUCK, M.: On a new family of generalized Stirling and Bell numbers, Electron. J. Combin. 18 (2011), #P77.10.37236/564Suche in Google Scholar

[15] MERRIS, R.: The p-Stirling numbers, Turkish J. Math. 24 (2000), 379–399.Suche in Google Scholar

[16] MEZÖ, I.: The r-Bell numbers, J. Integer Seq. 14 (2011), Art. ID 11.1.1.Suche in Google Scholar

[17] MIHOUBI, M. — RAHMANI, M.: The partial r-Bell polynomials, Afrika Mat. 28 (2017), 1167–1183.10.1007/s13370-017-0510-zSuche in Google Scholar

[18] NYUL, G. — RÁCZ, G.: The r-Lah numbers, Discrete Math. 338 (2015), 1660–1666.10.1016/j.disc.2014.03.029Suche in Google Scholar

[19] RAMÌREZ, J. L. — SHATTUCK, M.: A (p, q)-analogue of the r-Whitney-Lah numbers, J. Integer Seq. 19 (2016), Art. ID 16.5.6.Suche in Google Scholar

[20] SHATTUCK, M.: Generalized r-Lah numbers, Proc. Indian Acad. Sci. (Math Sci.) 126(4) (2016), 461–478.10.1007/s12044-016-0309-0Suche in Google Scholar

[21] SIMOVICI, D. A.: Several remarks on metrics on partition lattices and their applications in data mining, Libertas Math. 30 (2010), 19–31.Suche in Google Scholar

[22] SIMOVICI, D. A.: Entropies on bounded lattices. Proceedings of the 41st IEEE International Symposium on Multiple-Valued Logic, 2011, pp. 307–312.10.1109/ISMVL.2011.18Suche in Google Scholar

[23] SIMOVICI, D. A. — JAROSZEWICZ, S.: An axiomatization of partition entropy, IEEE Trans. Inform. Theory 48 (2002), 2138–2142.10.1109/TIT.2002.1013159Suche in Google Scholar

[24] SIMOVICI, D. A. — SIZOV, R.: On partition metric space, index function, and data compression, Sci. Ann. Comput. Sci. 28(1) (2018), 141–156.10.7561/SACS.2018.1.141Suche in Google Scholar

[25] WEISMAN, D. — SIMOVICI, D. A.: Several remarks on the metric space of genetic codes, Int. J. Data Mining and Bioinformatics 6(1) (2012), 17–26.10.1504/IJDMB.2012.045534Suche in Google Scholar

[26] ZHANG, P. — DEY, D.K.: The degree profile and Gini index of random caterpillar trees, Probab. Engrg. Inform. Sci. 33 (2019), 511–527.10.1017/S0269964818000475Suche in Google Scholar
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

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Gini index; set partition; Lah distribution; combinatorial statistic

Artikel in diesem Heft

  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
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 30.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2022-0077/html?lang=de
Button zum nach oben scrollen