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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
DOI: https://doi.org/10.1090/S0002-9939-1987-0911049-8
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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DOI: https://doi.org/10.1090/S0002-9939-1987-0911049-8
Keywords: Unimodal, Hall-Littlewood symmetric function, Kostka polynomial, symmetric chain order, Sperner property
Article copyright: © Copyright 1987 American Mathematical Society

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