6 research outputs found
On Cobweb Admissible Sequences - The Production Theorem
In this note further clue decisive observations on cobweb admissible
sequences are shared with the audience. In particular an announced proof of the
Theorem 1 (by Dziemia\'nczuk) from [1] announced in India -Kolkata- December
2007 is delivered here. Namely here and there we claim that any cobweb
admissible sequence F is at the point product of primary cobweb admissible
sequences taking values one and/or certain power of an appropriate primary
number p.
Here also an algorithm to produce the family of all cobweb-admissible
sequences i.e. the Problem 1 from [1] i.e. one of several problems posed in
source papers [2,3] is solved using the idea and methods implicitly present
already in [4]Comment: 6 page
On multi F-nomial coefficients and Inversion formula for F-nomial coefficients
In response to [6], we discover the looked for inversion formula for F-nomial
coefficients. Before supplying its proof, we generalize F-nomial coefficients
to multi F-nomial coefficients and we give their combinatorial interpretation
in cobweb posets language, as the number of maximal-disjoint blocks of the form
sP_{k_1,k_2,...,k_s} of layer Phi_n>. Then we present inversion
formula for F-nomial coefficients using multi F-nomial coefficients for all
cobweb-admissible sequences. To this end we infer also some identities as
conclusions of that inversion formula for the case of binomial, Gaussian and
Fibonomial coefficients.Comment: 11 pages, 2 figure
On Cobweb Posets and Discrete F-Boxes Tilings
F-boxes defined in [6] as hyper-boxes in N^{\infty} discrete space were
applied here for the geometric description of the cobweb posetes Hasse diagrams
tilings. The F-boxes edges sizes are taken to be values of terms of natural
numbers' valued sequence F. The problem of partitions of hyper-boxes
represented by graphs into blocks of special form is considered and these are
to be called F-tilings. The proof of such tilings' existence for certain
sub-family of admissible sequences F is delivered. The family of F-tilings
which we consider here includes among others F = Natural numbers, Fibonacci
numbers, Gaussian integers with their corresponding F-nomial (Binomial,
Fibonomial, Gaussian) coefficients. Extension of this tiling problem onto the
general case multi F-nomial coefficients is here proposed. Reformulation of the
present cobweb tiling problem into a clique problem of a graph specially
invented for that purpose - is proposed here too. To this end we illustrate the
area of our reconnaissance by means of the Venn type map of various cobweb
sequences families.Comment: 24 pages, 15 figures, Affiliated to The Internet Gian-Carlo Polish
Seminar http://ii.uwb.edu.pl/akk/sem/sem_rota.ht
Fibonomial cumulative connection constants
In this note we present examples of cumulative connection constants included
new fibonomial ones. All examples posses combinatorial interpretation.Comment: affiliated to The Internet Gian-Carlo Polish Seminar:
http://ii.uwb.edu.pl/akk/sem/sem_rota.ht
Counting Bipartite, k-Colored and Directed Acyclic Multi Graphs Through F-nomial coefficients
F-nomial coefficients encompass among others well-known binomial coefficients
or Gaussian coefficients that count subsets of finite set and subspaces of
finite vector space respectively. Here, the so called F-cobweb tiling sequences
N(a) are considered. For such specific sequences a new interpretation with
respect to Kwasniewski general combinatorial interpretation of F-nomial
coefficients is unearhed.
Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to
the number of labeled special bipartite multigraphs denoted here as
a-multigraphs G(a,n,k).
An explicit relation between the number of k-colored a-multigraphs and multi
N(a)-nomial coefficients is established. We also prove that the unsigned values
of the first row of inversion matrix for N(a) -nomial coefficients considered
here are equal to the numbers of directed acyclic a-multigraphs with n nodes.Comment: 11 pages, Affiliated to The Internet Gian-Carlo Polish Seminar
http://ii.uwb.edu.pl/akk/sem/sem_rota.ht
On Cobweb Admissible Sequences - The Production Theorem
In this note further clue decisive observations on cobweb admissible sequences are shared with the audience. In particular an announced proof of the Theorem 1 (by DziemiaĆczuk) from [1] announced in India-Kolkata-December 2007 is delivered here. Namely here and there we claim that any cobweb admissible sequence F is at the point product of primary cobweb admissible sequences taking values one and/or certain power of an appropriate primary number p. Here also an algorithm to produce the family of all cobweb-admissible sequences i.e. the Problem 1 from [1] i.e. one of several problems posed in source papers [2, 3] is solved using the idea and methods implicitly present already in [4]. Presented at Gian-Carlo Polish Seminar