Estimates on the number of partially ordered sets

TL;DR

This paper provides upper bounds on the number of certain partially ordered sets and permutations, explores their connections with combinatorial structures, and presents a generating function for related permutation counts.

Contribution

It introduces new bounds for counts of (k, n)-type posets and permutations, and surveys their relationships with various combinatorial objects.

Findings

Proved upper bounds for ppa_k(n) and si_k(n).

Connected posets, Young diagrams, and matrices through a comprehensive survey.

Presented the generating function for si_k(n) from Gessen's 1990 work.

Abstract

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.