Cobweb posets - Recent Results

TL;DR

This paper discusses recent progress in solving computational problems related to cobweb posets, a class of directed acyclic graphs with combinatorial significance, including their relation to Fibonacci sequences and bipartite chains.

Contribution

It reports on advancements in solving key computational problems for cobweb posets, enhancing understanding of their structure and combinatorial properties.

Findings

Progress in solving computational problems for cobweb posets

Identification of cobweb posets with chains of di-bicliques

Enhanced combinatorial interpretation of cobweb posets

Abstract

Cobweb posets uniquely represented by directed acyclic graphs are such a generalization of the Fibonacci tree that allows joint combinatorial interpretation for all of them under admissibility condition. This interpretation was derived in the source papers ([6,7] and references therein to the first author).[7,6,8] include natural enquires to be reported on here. The purpose of this presentation is to report on the progress in solving computational problems which are quite easily formulated for the new class of directed acyclic graphs interpreted as Hasse diagrams. The problems posed there and not yet all solved completely are of crucial importance for the vast class of new partially ordered sets with joint combinatorial interpretation. These so called cobweb posets - are relatives of Fibonacci tree and are labeled by specific number sequences - natural numbers sequence and Fibonacci sequence included. The cobweb posets might be identified with a chain of di-bicliques i.e. by definition - a chain of complete bipartite one direction digraphs [6]. Any chain of relations is therefore obtainable from the cobweb poset chain of complete relations via deleting arcs in di-bicliques of the complete relations chain. In particular we response to one of those problems [1].