On toral posets and contact Lie algebras

A $(2k+1)-$dimensional Lie algebra is called contact if it admits a one-form $\phi$ such that $\phi \wedge (d\phi {)}^k\mathrm{\ne }0.$ Here, we extend recent work to describe a combinatorial procedure for generating contact, type-A Lie poset algebras whose associated posets have chains of arbitrary cardinality, and we conjecture that our construction leads to a complete characterization.