First Observations on Prefab Posets’ Whitney Numbers

Abstract

We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers' triangular array -are then calculated and the explicit formula for them is provided. Next -in the second construction -we endow the set sums of prefabiants with such an another partial order that their Bell-like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial-like coefficients (incidence coefficients included). The first partial order is F -sequence independent while the second partial order is F -sequence dependent where F is the so-called admissible sequence determining cobweb poset by construction. An F -determined cobweb poset's Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so-called extended coherent states of quantum physics. This information is delivered in the last section. (2000). 05C20, 11C08, 17B56. Mathematics Subject Classification

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: [6,7]). 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 [8,9]).…”

Cobweb posets - Recent Results

“…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.…”

“…This report is based on [6][7] from which definitions and description of these new DAG's are quoted for the sake of self consistency and last results are taken from [1]. Applications of new cobweb posets' originated Whitney numbers from [8] such as extended Stirling or Bell numbers are expected to be of at least such a significance in applications to linear algebra of formal series as Stirling numbers, Bell numbers or their q-extended correspondent already are in the so called coherent states physics [9,10] (see [13] for abundant references on this subject and other applications such as in [11,12]).…”

“…This is being now under investigation with a progress to be announced soon by Ewa Krot-Sieniawska [4,5] -the member of our Rota Polish Seminar Group. The recent papers on DAGs related to this article and its clue references [6,7,8] apart from [1] are [4] and [14]. Computation and Characterizing Problems.…”

“…In the next section we define cobweb posets. Their examples are given [6][7][8]1]. A join combinatorial interpretation of cobweb posets' characteristic binomiallike coefficients is provided too following the source papers of the first author.…”