Antichains of interval orders and semiorders, and Dilworth lattices of maximum size antichains

Abstract

This thesis consists of two parts. In the first part we count antichains of interval orders and in particular semiorders. We associate a Dyck path to each interval order, and give a formula for the number of antichains of an interval order in terms of the corresponding Dyck path. We then use this formula to give a generating function for the total number of antichains of semiorders, enumerated by the sizes of the semiorders and the antichains. In the second part we expand the work of Liu and Stanley on Dilworth lattices. Let L be a distributive lattice, let -(L) be the maximum number of elements covered by a single element in L, and let K(L) be the subposet of L consisting of the elements that cover o-(L) elements. By a result of Dilworth, K(L) is also a distributive lattice. We compute o(L) and K(L) for various lattices L that arise as the coordinate-wise partial ordering on certain sets of semistandard Young tableaux.