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Enumeration of posets generated by disjoint unions and ordinal sums


Author: Richard P. Stanley
Journal: Proc. Amer. Math. Soc. 45 (1974), 295-299
MSC: Primary 06A10
DOI: https://doi.org/10.1090/S0002-9939-1974-0351928-7
MathSciNet review: 0351928
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Abstract: Let $ {f_n}$ be the number of $ n$-element posets which can be built up from a given collection of finite posets using the operations of disjoint union and ordinal sum. A curious functional equation is obtained for the generating function $ \Sigma {f_n}{x^n}$. Using a result of Bender, an asymptotic estimate can sometimes be given for $ {f_n}$. The analogous problem for labeled posets is also considered.


References [Enhancements On Off] (What's this?)

  • [1] E. A. Bender, Asymptotic methods in enumeration, SIAM Rev. (to appear). MR 0376369 (51:12545)
  • [2] D. Kleitman and B. Rothschild, The number of finite topologies, Proc. Amer. Math. Soc. 25 (1970), 276-282. MR 40 #7157. MR 0253944 (40:7157)
  • [3] J. Riordan, An introduction to combinatorial analysis, Wiley, New York; Chapman & Hall, London, 1958. MR 20 #3077. MR 0096594 (20:3077)
  • [4] J. Wright, Cycle indices of certain classes of quasiorder types or topologies, Dissertation, University of Rochester, Rochester, N. Y., 1972.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0351928-7
Keywords: Poset, partially ordered set, disjoint union, ordinal sum, generating function, Pólya's enumeration theorem, functional equation
Article copyright: © Copyright 1974 American Mathematical Society

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