Computational and Combinatorial Enumeration of Poset Matrices

This paper is a sequel of arXiv:2401.06814; here a computational algorithm is presented to enumerate poset matrices generated from their operad and partial composition operations. The results of this enumeration procedure leads to the first known connections of poset matrices to the combinatorial sequences representing the large Schroeder numbers, super-Catalan numbers, and Fibonaccinumbers. In addition, a set of efficient algorithms are introduced for the classification of poset matrices in terms of their different forms of duality, connectivity, and series-parallelism. Poset matrices generated from operad and partial composition operations are also enumerated based on this classification. An open source Python implementation of all the algorithms described in this paper is provided.