Cobweb posets - Recent Results

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].

27 pages, 15 figures

References in corpus (14)

• First Observations on Prefab Posets Whitney Numbers
• On Cobweb posets tiling problem
• On incidence algebras description of cobweb posets
• On Cobweb Posets and Discrete F-Boxes Tilings
• On cobweb posets most relevant codings
• Some Cobweb Posets Digraphs' Elementary Properties and Questions
• On multi F-nomial coefficients and Inversion formula for F-nomial coefficients
• On GCD-morphic sequences
• Graded posets zeta matrix formula
• Reduced Incidence algebras description of cobweb posets and KoDAGs
• How the work of Gian Carlo Rota had influenced my group research and life
• On inversion formulas and Fibonomial coefficients
• Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices
• On duality triads

Cited by in corpus (18)

• On incidence algebras description of cobweb posets
• On Cobweb Posets and Discrete F-Boxes Tilings
• On cobweb posets most relevant codings
• On multi F-nomial coefficients and Inversion formula for F-nomial coefficients
• On GCD-morphic sequences
• Reduced Incidence algebras description of cobweb posets and KoDAGs
• How the work of Gian Carlo Rota had influenced my group research and life
• On Characteristic Polynomials of the Family of Cobweb Posets
• Graded posets inverse zeta matrix formula
• Counting Bipartite, k-Colored and Directed Acyclic Multi Graphs Through F-nomial coefficients
• Characterization of Cobweb Posets as KoDAGs
• On inversion formulas and Fibonomial coefficients
• On natural join of posets properties and first applications
• Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices
• Natural join construction of graded posets versus ordinal sum and discrete hyper boxes
• Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
• New formulas for Stirling-like numbers and Dobinski-like formulas
• On Cobweb Admissible Sequences - The Production Theorem