Minimal Posets with Prescribed Maximal Chain Cardinalities

Todd Bichoupan

paper

Minimal Posets with Prescribed Maximal Chain Cardinalities

arXiv:2305.17541

Abstract

Given a nonempty finite multiset $S$ of positive integers, we wish to find a partially ordered set $P$ of minimal cardinality such that the multiset of cardinalities of all maximal chains in $P$ equals $S$ . This paper establishes upper and lower bounds on the size of $P$ : $max (S) + ⌈ lo g_{2} ∣ S ∣ ⌉ <= ∣ P ∣ <= max (S) + ∣ S ∣ - 1$ , and both bounds are tight.