On the average rank of LYM-sets (Q1898339)

If \(S\) is a finite set with some rank with log-concave Whitney numbers satisfying \(w_{k - 1} < w_k \leq w_{k + m}\), \(W = w_k + w_{k + 1} + \cdots + w_{k + m}\), \(F\) is a subset of \(S\) of cardinality at least \(W\) with a LYM-type normalized profile vector then the average rank of \(F\) is at least \((kw_k + \cdots + (k + m) w_{k + m})/W\). This generalizes a theorem of D. J. Kleitman and E. C. Milner. An inequality on log-concave sequences of positive reals is also established.