The estimate of the number of permutationally-ordered sets

Abstract

It is proved that the number of n-element permutationally-ordered sets with the maximal anti-chain of length k is not greater than \(\min \left\{ {\tfrac{{k^{2n} }} {{(k!)^2 }},\tfrac{{(n - k + 1)^{2n} }} {{((n - k)!)^2 }}} \right\}\). It is also proved that the number of permutations £k(n) of the numbers {1,..., n} with the maximal decreasing subsequence of length at most k satisfies the inequality \(\tfrac{{k^{2n} }} {{((k - 1)!)^2 }}\). A review of papers focused on bijections and relations between pairs of linear orders, pairs of Young diagrams, two-dimensional integer arrays, and integer matrices is presented.