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Partition Lattice with Limited Block Sizes

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Abstract

In this paper we study two modifications of the partition lattice, regarding the size of the blocks. After determining the most elementary properties of these lattices, we calculate their Möbius functions, and study the enumeration problems of the chains and maximal length chains in our lattices.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. There are partitions which cover \(\hat{0}\), but does not satisfy this maximality condition. Take, for example, \(1\,2\,3 | 4\,5\,6\in \mathcal {P}_{\ge 2}(6)\). This element covers \(\hat{0}\), but it contains less than the maximal \(6/2=3\) number of blocks, so a chain \(\hat{0}<1\,2\,3|4\,5\,6<\cdots \) cannot be maximal.

References

  1. Aigner, M.: Combinatorial Theory. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  2. Barsky, D., Bézivin, J.-P.: \(p\)-adic properties of Lengyel’s numbers. J. Integer Seq. 17 (2014) (Article 14.7.3)

  3. Bényi, B., Ramírez, J.L.: Some applications of \(S\)-restricted set partitions. Period. Math. Hungar. 78, 110–127 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Björner, A., Sagan, B.E.: Subspace arrangements of type \(B_n\) and \(D_n\). J. Algebraic Combin. 5, 291–314 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Björner, A., Wachs, M.L.: Shellable nonpure complexes and posets I. Trans. Am. Math. Soc. 348(4), 1299–1327 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caicedo, J. B., Moll, V. H., Ramírez, J. L., Villamizar, D.: Extensions of set partitions and permutations. Electron. J. Combin. 26(2) (2019) (P2.20)

  7. Ehrenborg, R., Readdy, M.A.: Exponential Dowling structures. Eur. J. Combin. 30, 311–326 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Engbers, J., Galvin, D., Smyth, C.: Restricted Stirling and Lah numbers and their inverses. J. Combin. Theory Ser. A 161, 271–298 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Erdős, P., Guy, R.K., Moon, J.W.: On refining partitions. J. Lond. Math. Soc. 9(2), 565–570 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gould, H. W., Kwong, H., Quaintance, J.: On certain sums of Stirling numbers with binomial coefficients. J. Integer Seq. 18 (2015) (Article 15.9.6)

  11. Grätzer, G.: Lattice Theory: Foundation. Birkhäuser (2011)

    Book  MATH  Google Scholar 

  12. Komatsu, T., Liptai, K., Mező, I.: Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers. Publ. Math. Debrecen 88, 357–368 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lai, N. J. H.: \(p\)-adic properties of recurrences involving Stirling numbers, manuscript (2014)

  14. Lengyel, T.: On a recurrence involving Stirling numbers. Eur. J. Combin. 5, 313–321 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lengyel, T.: On some 2-adic properties of a recurrence involving Stirling numbers. p-Adic Numbers. Ultrametric Anal. Appl. 4(3), 179–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Linusson, S.: Partitions with restricted block sizes, Möbius functions, and the \(k\)-of-each problem. SIAM J. Discrete Math. 10(1), 18–29 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mansour, T., Shattuck, M.: A generalized class of restricted Stirling and Lah numbers. Math. Slovaca 68(4), 727–740 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mansour, T., Shattuck, M.: A polynomial generalization of some associated sequences related to set partitions. Period. Math. Hungar. 75, 398–412 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mező, I.: Combinatorics and Number Theory of Counting Sequences. Chapman & Hall/CRC (2019)

    MATH  Google Scholar 

  20. Mező, I.: Periodicity of the last digits of some combinatorial sequences. J. Integer Seq.17 (2014) (article 14.1.1)

  21. Miksa, F.L., Moser, L., Wyman, M.: Restricted partitions of finite sets. Canad. Math. Bull. 1, 87–96 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  22. Moll, V., Ramírez, J.L., Villamizar, D.: Combinatorial and arithmetical properties of the restricted and associated Bell and factorial numbers. J. Comb. 9(4), 693–720 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Sloane, N. J. A.: The On-line Encyclopedia of Integer Sequences, available electronically at http://oeis.org

  24. Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Univ Press (1997)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the referees for the detailed comments and corrections that help us to improve the paper.

Funding

The third author was partially supported by Universidad Nacional de Colombia, Project No. 53490.

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Authors and Affiliations

  1. Instituto de Matemática y Física, Universidad de Talca, Campus Norte, Camino Lircay S/N, Talca, Chile

    Jhon B. Caicedo

  2. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, No. 219 Ningliu Rd., Nanjing, People’s Republic of China

    István Mező

  3. Institute of Applied Mathematics, John von Neumann Faculty of Informatics, Óbuda University, Budapest, Hungary

    István Mező

  4. Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia

    José L. Ramírez

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  1. Jhon B. Caicedo
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  2. István Mező
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  3. José L. Ramírez
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Correspondence to István Mező.

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Caicedo, J.B., Mező, I. & Ramírez, J.L. Partition Lattice with Limited Block Sizes. Graphs and Combinatorics 38, 146 (2022). https://doi.org/10.1007/s00373-022-02548-1

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  • DOI: https://doi.org/10.1007/s00373-022-02548-1

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