How Often Are Two Permutations Comparable?
نویسنده
چکیده
Two permutations of [n] are comparable in the Bruhat order if one is closer, in a natural way, to the identity permutation, 1 2 · · · n, than the other. We show that the number of comparable pairs is of order (n!) /n2 at most, and (n!) (0.708) at least. For the related weak order, the corresponding bounds are (n!) (0.362) and (n!) ∏n i=1 (H (i) /i), where H (i) := ∑i j=1 1/j. In light of numerical experiments, we conjecture that for each order the upper bound is qualitatively close to the actual number of comparable pairs.
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تاریخ انتشار 2008