On the Average Rank of LYM-sets

Erdős, Péter and Faigle, U. and Kern, W. (1995) On the Average Rank of LYM-sets. DISCRETE MATHEMATICS, 144 (1-3). pp. 11-22. ISSN 0012-365X

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Abstract

Let S be a finite set with some rank function r such that the Whitney numbers w(i) = \{x is an element of S\r(x)= i}\ are log-concave. Given k, m is an element of N so that w(k-1) < w(k) less than or equal to w(k+m), set W = w(k) + w(k+1) + ... + w(k+m). Generalizing a theorem of Kleitman and Milner, we prove that every F subset of or equal to S with cardinality \F\ greater than or equal to W has average rank at least (kw(k) + ... + (k + m)w(k+m))/W, provided the normalized profile vector (x(1),...,x(n)) of F satisfies the following LYM-type inequality: x(0) + x(1) + ... + x(n) less than or equal to m + 1.

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