arXiv subjects

A. Krzysztof Kwaśniewski

Publications and source records attributed to A. Krzysztof Kwaśniewski.

3 recordsLinked to original sources

arXiv2009

On natural join of posets properties and first applications

In addition to the three standard operations on posets which are dual of poset or ordinal and cardinal sums of partial ordered sets one adds the natural join of posets. This is especially natural natural join operation for graded posets including very special and important the so called cobweb posets. The main aims of this article are the presentation of the authors update applications of natural join of posets in the domain of graded posets and summary of the general properties of natural join of posets with posing some questions arising on the way. Thus in this note apart from revealing some general properties of natural join of posets we also deliver the authors combinatorial interpretations of cobweb posets including discrete hyperboxes coding of the so called cobweb posets .Various explicit formulas for the zeta functions and zeta matrices as well as the inverse of zeta matrix for any graded posets are supplied. Whitney numbers formulas and those for characteristic polynomials are given too.

math.CO↗

arXiv2008

On inversion formulas and Fibonomial coefficients

A research problem for undergraduates and graduates is being posed as a cap for the prior antecedent regular discrete mathematics exercises. [Here cap is not necessarily CAP=Competitive Access Provider, though nevertheless ...] The object of the cap problem of final interest i.e. array of fibonomial coefficients and the issue of its combinatorial meaning is to be found in A.K.Kwaśniewski's source papers. The cap problem number seven - still opened for students has been placed on Mathemagics page of the first author [http://ii.uwb.edu.pl/akk/dydaktyka/dyskr/dyskretna.htm]. The indicatory references are to point at a part of the vast domain of the foundations of computer science in ArXiv affiliation noted as CO.cs.DM. The presentation has been verified in a tutor system of communication with a couple of intelligent students. The result is top secret.Temporarily. [Contact: Wikipedia; Theory of cognitive development].

math.CO↗

arXiv2008

First Observations on Prefab Posets Whitney Numbers

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 posets 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.

math.CO↗