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 thereand 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. One presents here also a join combinatorial interpretation of those posets' F -nomial coefficients which are computed with the so called cobweb admissible sequences. Cobweb posets and their natural subposets are graded posets. They are vertex partitioned into such antichains Φn (where n is a nonnegative integer) that for each Φn, all of the elements covering x are in Φn+1 and all the elements covered by x are in Φn. We shall call the Φn the n − th-level. 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]. This is a tiling problem. Our information on tiling problem refers on proofs of tiling's existence for some cobweb-admissible sequences as in [1]. There the second author shows that not all cobwebs admit tiling as defined below and provides examples of cobwebs admitting tiling.

On multi F-nomial coefficients and Inversion formula for F-nomial coefficients

“…In this note we derive inversion formula, not only for Fibonacci number, but for all F-cobweb admissible sequences [4,5,6,10]. Therefore we can expect general hence simpler form of new and known identities for certain sequences, such as for example for Natural and Gaussian numbers as we shall present it further on.…”

Reduced incidence algebras description of cobweb posets and KoDAGs

“…The family of the so called cobweb posets Π has been invented by A.K.Kwaśniewski few years ago (for references see: [8,9]). These structures are such a generalization of the Fibonacci tree growth that allows joint combinatorial interpretation for all of them under the admissibility condition (see [10,11]). Let {F n } n≥0 be a natural numbers valued sequence with F 0 = 1 (with F 0 = 0 being exceptional as in case of Fibonacci numbers).…”

“…The Kwasniewski cobweb posets under consideration represented by graphs are examples of oderable directed acyclic graphs (oDAG) which we start to call from now in brief: KoDAGs. These are structures of universal importance for the whole of mathematics -in particular for discrete "'mathemagics"' [http://ii.uwb.edu.pl/akk/ ] and computer sciences in general (quotation from [10,11] ):…”

Graded posets inverse zeta matrix formula

“…F n ≡ n F . The Upside Down Notation Principle used since last century effectively (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and for earlier references therein; in particular see Appendix in [9] copied from [32]) may be formulated as a Principle i.e. trivial, powerful statement as follows.…”