Order

pp 1–7 | Cite as

Counterexamples to Conjectures About Subset Takeaway and Counting Linear Extensions of a Boolean Lattice

Article

Abstract

We develop an algorithm for efficiently computing recursively defined functions on posets. We illustrate this algorithm by disproving conjectures about the game Subset Takeaway (Chomp on a hypercube) and computing the number of linear extensions of the lattice of a 7-cube and related lattices.

Keywords

Combinatorial game Impartial game Grundy number Partially ordered set Total order 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brightwell, G.R., Tetali, P.: The number of linear extensions of the Boolean lattice. Order 20, 333–345 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brightwell, G.R., Winkler, P.: Counting linear extensions. Order 8, 225–242 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
  4. 4.
    Christensen, J.D., Tilford, M.: David Gale’s subset take-away game. Amer. Math. Monthly 104, 762–766 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Conway, J.H.: On Numbers and Games. Academic Press, London (1976)MATHGoogle Scholar
  6. 6.
    Gale, D., Neyman, A.: Nim-type games. Internat. J. Game Theory 11, 17–20 (1982)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gardner, M.: Mathematical games, Sci. Am., 1973, pp. 110–111Google Scholar
  8. 8.
    Khandhawit, T., Ye, L.: Chomp on Graphs and Subsets. arXiv:1101.2718 (2011)
  9. 9.
    Morton, J.R.: Geometry of conditional independence, Ph.D. thesis, University of California, Berkeley, Fall (2007)Google Scholar
  10. 10.
    Sha, J., Kleitman, D.J.: The number of linear extensions of subset ordering, vol. 63 (1987)Google Scholar
  11. 11.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. A000372, A003182 and A046873
  12. 12.
    Weinreich, D.M.: The rank ordering of genotypic fitness values predicts genetic constraint on natural selection on landscapes lacking sign epistasis. Genetics 171, 1397–1405 (2005)CrossRefGoogle Scholar
  13. 13.
    Wiedemann, D.H.: A computation of the eighth Dedekind number. Order 8, 5–6 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wienand, O.: Algorithms for Symbolic Computation and Their Applications, Ph.D. thesis, Univ. Kaiserslautern (2011)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada

Personalised recommendations