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On the average rank of LYM-sets

Authors
Journal
Discrete Mathematics
0012-365X
Publisher
Elsevier
Publication Date
Volume
144
Identifiers
DOI: 10.1016/0012-365x(94)00282-n
Disciplines
  • Mathematics

Abstract

Abstract Let S be a finite set with some rank function r such that the Whitney numbers w i = |{ x ∈ S| r( x) = i}| are log-concave. Given k, N so that w k − 1 < w k ⩽ 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 ⊆ S with cardinality | F| ⩾ 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 ⩽ m + 1.

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