Elsevier

Discrete Mathematics

Volume 144, Issues 1–3, 8 September 1995, Pages 11-22
Discrete Mathematics

On the average rank of LYM-sets

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Abstract

Let S be a finite set with some rank function r such that the Whitney numbers wi = |{xS|r(x) = i}| are log-concave. Given k, N so that wk − 1 < wkwk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every FS with cardinality |F| ⩾ W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xnm + 1.

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