Counting canonical partitions in the random graph

Abstract

Joyce trees have concrete realizations as J-trees of sequences of 0’s and 1’s. Algorithms are given for computing the number of minimal height J-trees of d-ary sequences with n leaves and the number of them with minimal parent passing numbers to obtain polynomials ρ n (d) for the full collection and α n (d) for the subcollection.

The number of traditional Joyce trees is the tangent number α n (1); α n (2) is the number of cells in the canonical partition by Laflamme, Sauer and Vuksanovic of n-element subsets of the infinite random (Rado) graph; and ρ n (2) is the number of weak embedding types of rooted n-leaf J-trees of sequences of 0’s and 1’s.

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References

  1. [1]

    Denis C. Devlin: Some partition theorems and ultrafilters on ω, Ph.D. thesis, Dartmouth College, 1979.

  2. [2]

    Paul Erdős, András Hajnal and Lajos Pósa: Strong embeddings of graphs into colored graphs, Colloquia Mathematica Societatis János Bolyai, 10, Vol. I, Infinite and Finite Sets, 585–595.

  3. [3]

    Paul Erdős and Richard Rado: A combinatorial theorem, J. London Math. Soc. 25(2) (1950), 249–255.

    Article  MathSciNet  Google Scholar 

  4. [4]

    Ronald L. Graham, Donald K. Knuth and Oren Patashnik: Concrete Mathematics, Addison-Wesley Publishing Company, New York, 1989.

    MATH  Google Scholar 

  5. [5]

    Claude Laflamme, Norbert W. Sauer and Vojkan Vuksanovic: Canonical partitions of universal structures, Combinatorica 26(2) (2006), 183–205.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    Keith R. Milliken: A Ramsey theorem for trees, J. Combin. Theory Ser. A 26 (1979), 215–237.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    George N. Raney: Functional composition patterns and power series reversion, Trans. Amer. Math. Soc. 94 (1960), 441–451.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    Norbert W. Sauer: Edge partitions of the countable triangle free homogeneous graph, Discrete Mathematics 185 (1998), 137–181.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    Norbert W. Sauer: Appendix in: Theory of Relations, revised edition, by Roland Fraïssé, Studies in Logic and the Foundations of Mathematics, 145, North Holland, 2000.

  10. [10]

    Norbert W. Sauer: Coloring subgraphs of the Rado graph, Combinatorica 26(2) (2006), 231–253.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/:_njas/sequences/

  12. [12]

    Richard P. Stanley: Enumerative Combinatorics, Volume 2, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  13. [13]

    Ross Street: Trees, permutations and the tangent function; Reflections 27(2) (Math. Assoc. of New South Wales, May 2002), 19–23.

    MathSciNet  Google Scholar 

  14. [14]

    Vojkan Vuksanovic: A combinatorial proof of a partition relation for [ℚ]n, Proc. Amer. Math. Soc. 130(10) (2002), 2857–2864.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    Vojkan Vuksanovic: Canonical equivalence relations on ℚn, Order 20(4) (2003), 373–400.

    Article  MathSciNet  Google Scholar 

  16. [16]

    Vojkan Vuksanovic: Infinite partitions of random graphs, J. Combin. Theory Ser. A 113(2) (2006), 225–250.

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Jean A. Larson.

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The author thanks the University of Tel Aviv for hospitality in April 2004 when much of this work was done.

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Larson, J.A. Counting canonical partitions in the random graph. Combinatorica 28, 659–678 (2008). https://doi.org/10.1007/s00493-008-2148-9

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Mathematics Subject Classification (2000)

  • 05A15
  • 05C05
  • 03E02
  • 05C80