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A unimodality result in the enumeration of subgroups of a finite abelian group

Author: Lynne M. Butler
Journal: Proc. Amer. Math. Soc. 101 (1987), 771-775
MSC: Primary 05A15; Secondary 20D60, 20K01
MathSciNet review: 911049
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Abstract: The number of subgroups of order $ {p^k}$ in an abelian group $ G$ of order $ {p^n}$ is a polynomial in $ p,{\alpha _ \leftthreetimes }(k;p)$, determined by the type $ \lambda $ of $ G$. It is well known that $ {\alpha _ \leftthreetimes }(k;p) = {\alpha _ \leftthreetimes }(n - k;p)$. Using a recent result from the theory of Hall-Littlewood symmetric functions, we prove that $ {\alpha _ \leftthreetimes }(k;p)$, is a unimodal sequence of polynomials. That is, for $ 1 \leq k \leq n/2,{\alpha _\lambda }(k;p) - {\alpha _\lambda }(k - 1;p)$ is a polynomial in $ p$ with nonnegative coefficients.

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  • [1] M. Aigner, Combinatorial theory, Springer-Verlag, New York, 1979. MR 542445 (80h:05002)
  • [2] Garrett Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. (2) 38 (1934-35), 385-401.
  • [3] S. N. Bhatt and C. E. Leiserson, How to assemble tree machines, Proceedings of the 14th Symposium on Theory of Computing, San Francisco, May 5-7, 1982, pp. 77-84.
  • [4] A. Björner, A. M. Garsia, and R. P. Stanley, An introduction to Cohen-Macaulay partially ordered sets, Ordered Sets, edited by I. Rival, Reidel, Dordrecht and Boston, Mass., 1982, pp. 583-615. MR 661307 (83i:06001)
  • [5] L. M. Butler, Combinatorial properties of partially ordered sets associated with partitions and finite abelian groups, Ph.D. thesis, M. I. T., 1986.
  • [6] L. Carlitz, Seguences and inversions, Duke Math. J. 37 (1970), 193-198. MR 0252237 (40:5458)
  • [7] J. R. Griggs, On chains and Sperner $ k$-families in ranked posets, J. Combinatorial Theory A28 (1980), 156-168. MR 563553 (81e:05006)
  • [8] A. Lascoux and M.-P. Schützenberger, Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris 286A (1978), 323-324. MR 0472993 (57:12672)
  • [9] D. E. Littlewood, On certain symmetric functions, Proc. London Math. Soc. (3) 11 (1961), 485-498. MR 0130308 (24:A173)
  • [10] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979. MR 553598 (84g:05003)
  • [11] B. E. Sagan, Inductive and injective proofs of log-concavity results, preprint. MR 926131 (89b:05009)
  • [12] M.-P. Schützenberger, Propriétés nouvelles des tableaux de Young, Séminaire Delange-Pisot-Poitou, 19 $ ^{\emph{e}}$, 1977/78, no. 26, Secrétariat Mathématique, Paris. MR 520318 (80g:20023)
  • [13] R. Stanley, Some aspects of groups acting on finite posets, J. Combinatorial Theory A32 (1982), 132-161. MR 654618 (83d:06002)

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Keywords: Unimodal, Hall-Littlewood symmetric function, Kostka polynomial, symmetric chain order, Sperner property
Article copyright: © Copyright 1987 American Mathematical Society

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