Estimates on the number of partially ordered sets

Partially ordered sets of type (k, n) are the sets such that a) cardinality of each set is n, b) dimension of each set is two, c) length of the maximal antichain in each set is k. Let \alpha_k(n) be the number of partially ordered sets of type (k, n). We prove that \alpha_k(n)<min{k^{2n}/((k!)^2), (n-k+1)^{2n}/(((n-k)!)^2)}. Denote by \xi_k(n) the number of permutations from S_n such that the maximal decreasing chain of such permutation has length k. We prove that \xi_k(n)<k^{2n}/(((k-1)!)^2). We survey connections among the pairs of linear orders, the pairs Young diagrams, two-dimensional arrays of positive integers and matrices of nonnegative integers. This survey is based on papers of Schensted and Knuth. We show the generating function of \xi_k(n). It was obtained by Gessen in 1990.