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Two by two squares in set partitions

  • Margaret Archibald , Aubrey Blecher , Charlotte Brennan , Arnold Knopfmacher and Toufik Mansour
Published/Copyright: January 13, 2020
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 1

Abstract

A partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that ⋃i=1kBi=S. We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and ⋃i=1n{πi}=[k], and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].

MSC 2010: Primary 05A15
Keywords: set partitions; restricted growth functions; generating functions; Bell numbers

The first, third and fourth authors are supported by the National Research Foundation under grant numbers 89147, 86329 and 81021, respectively.

  1. Communicated by Anatolij Dvurečenskij

References

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Received: 2018-07-20
Accepted: 2019-07-23
Published Online: 2020-01-13
Published in Print: 2020-02-25

© 2020 Mathematical Institute Slovak Academy of Sciences

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  2. Doc. RNDr. Roman Frič, DrSc. – 3/4 C?
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  5. Identifying quantum logics by numerical events
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  9. Maximum term of transcendental entire function and spider’s web
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  12. Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator
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  17. On 4-th root metrics of isotropic scalar curvature
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https://doi.org/10.1515/ms-2017-0328
Keywords for this article
set partitions; restricted growth functions; generating functions; Bell numbers

Articles in the same Issue

  1. Regular papers
  2. Doc. RNDr. Roman Frič, DrSc. – 3/4 C?
  3. Sugihara algebras and Sugihara monoids: Multisorted dualities
  4. Two by two squares in set partitions
  5. Identifying quantum logics by numerical events
  6. Some remarks on divisible polyhedral MV-algebras
  7. Remarks on b-metrics, ultrametrics, and metric-preserving functions
  8. Estimates of essential norm of generalized weighted composition operators from Bloch type spaces to nth weighted type spaces
  9. Maximum term of transcendental entire function and spider’s web
  10. Entire solutions of certain type of non-linear differential equations
  11. On the oscillatory behavior of third order differential equations with a sublinear neutral term
  12. Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator
  13. The amalgam space L(p,q)π(G) on IN-groups
  14. On the topology of partial metric spaces
  15. The trace of the commutator of the Cesàro operator and a compact operator
  16. Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold
  17. On 4-th root metrics of isotropic scalar curvature
  18. On ultrametric-preserving functions
  19. Porosity of 𝓢-approximately continuous functions
  20. On a modified Burr XII distribution having flexible hazard rate shapes
  21. A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums
  22. An alternative estimate for the numerical radius of Hilbert space operators
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