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Mathematics » Cobweb posets

(1) Cobweb-posets, KoDAGs

$Internet Gian Carlo Rota Polish Seminar$

$Cobweb Poset of Fibonacci Numbers$
Fig.1 Cobweb Poset of Fibonacci Numbers

Cobweb Sequences and its Pascal-like triangles:

List of Cobweb Sequences: Admissible, Tileable and others.

Cobweb Poset Gallery:

$Cobweb Poset Gallery$
Gallery of cobweb poset structures and other relevant pictures.

Cobweb Gallery

Cobweb multi-blocks:

$KoDAGs multi-blocks$ This applet draws all multi-blocks of the form σP_k,n-k of layer <Φ₁ → Φ_n>. These blocks contain two blocks σP_k (red points) and σP_n-k (blue points). We can imagine them as a subsets X of N, such that red points belong to subset X and blue points don't.

Go to KoDAGs multi-tiling presenter

Cobweb posets tiling process:

$Drawing Cobweb Poset's applet$ This applet draws a Hasse diagram of Cobweb sub-poset levels and generates the partition of any layer <Φ_k → Φ_n> for the Natural numbers and the Fibonacci numbers. There are more different partitions of certain layer, but we know that exists at least one and this applet generates and draws them.

Go to Cobweb Poset's tiling applet

Cobweb Poset's drawing applet:

$Drawing Cobweb Poset's applet$ This applet draws finite Cobweb sub-posets defined by some sequences like the Natural numbers, the Fibonacci numbers and others.

In future I'm going to publish here the applet which will show certain partitioned layer with help of max-disjoint blocks of the form σP_m (Tiling Problem from [1, 2])

Go to Cobweb Poset's drawing applet

Simple examples of Cobweb Poset tiling:

$Cobweb Poset tiling$ $Cobweb Poset Tiling$

Cobweb Tiling Phenomenon in Geometric Interpretation:

$Cobweb Tiling - Geometric interpretation$
Fig.2 Cobweb Tiling using Geometric Interpretation. See [md6]

(2) KoDAGs problems (Open Problems)

A few of un-solved (still?) problems concern cobweb posets (KoDAGs):

Tiling problem - Find formula for the number of tilings of certain layer (see [md3,md4])
Tiling problem - Is Tiling problem a NP-complete problem? (see [md3,md4])
Tiling sequences - Define family of All cobweb-tiling sequences (see [md3,md4])
Incidence Algebra - Is there a poset, such that F-nomial coefficients are its Whitney numbers of second kind? (Prof.A.K.Kwasniewski problem [???])

If you are interested to solve one of them and need some additional information, do not hesitate to contact me.

(3) In memory of Andrzej Krzysztof Kwaśniewski

Andrzej Krzysztof Kwaśniewski (1947-2011)

(4) Preprints, publications

[md1] M. Dziemiańczuk, W.Bajguz, On GCD-morphic sequences, IeJNART: Volume (3), September 2009, 33-37.
[md2] M. Dziemiańczuk, On Cobweb Admissible Sequences - The Production Theorem, in Proceedings of The 2008 International Conference on Foundations of Computer Science (FCS'08), Interesting results, new models, and methodologies, July 14-17, 2008, Las Vegas, USA pp.163-165
[md3] A. Krzysztof Kwasniewski, M. Dziemianczuk, Cobweb posets - Recent Results, ISRAMA 2007, December 1-17 2007 Kolkata, INDIA, Adv. Stud. Contemp. Math. volume 16 (2), 2008 (April) pp. 197-218.
[md4] M. Dziemianczuk, On Cobweb posets tiling problem, Adv. Stud. Contemp. Math. volume 16 (2), 2008 (April) pp. 219-233.
[md5] M. Dziemianczuk, On Cobweb Posets and Discrete F-Boxes Tilings, ArXiv:0802.3473, 2 Apr 2009
[md6] A. Krzysztof Kwasniewski, M. Dziemianczuk, On cobweb posets most relevant codings, Preprint arXiv:0804.1728, 27 Feb 2009
[md7] M. Dziemianczuk, On multi F-nomial coefficients and Inversion formula for F-nomial coefficients, Preprint arXiv:0806.3626, 23 Jun 2008
[md8] M. Dziemianczuk, Counting Bipartite, k-Colored and Directed Acyclic Multi Graphs Through F-nomial coefficients, Preprint arXiv:0901.1337, 11 Jan 2009
[md9] M. Dziemianczuk, Generalization of Fibonomial Coefficients, Preprint arXiv:0908.3248, 22 Aug 2009

(5) Additionals pages about Cobweb Posets

Cobweb posets sequences map - map of Cobweb Admissible sequences, Cobweb Tiling and GCD-morphic sequences.
Cobweb poset's drawing applet - applet in java which draws the Cobweb poset' Hasse diagram
On tiling method - on the method of partitions production of any layer for the Natural and Fibonacci numbers. This method gives us exactly one partition of certain layer.

(6) Source materials and references

Gian Carlo Rota Polish Seminar Publications

[akk1] A. Krzysztof Kwaśniewski, On cobweb posets and their combinatorially admissible sequences Adv. Studies Contemp. Math. Vol. 18 No 1, 2009 17-32.
[akk2] A.K.Kwaśniewski, Cobweb posets as noncommutative prefabs, Adv. Stud. Contemp. Math. vol.14 (1) 2007. pp. 37-47; cs.DM ArXiv: math.CO/0503286 PS
[akk3] A.K.Kwaśniewski, Cobweb posets as noncommutative prefabs, Adv. Stud. Contemp. Math. vol.14 (1) 2007. pp. 37-47; cs.DM ArXiv: math.CO/0503286 PS
[akk4] A. Krzysztof Kwaśniewski, Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices ArXiv:0812.4066 Sun, 21 Dec 2008
[akk4] A. Krzysztof Kwaśniewski, Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices ArXiv:0812.4066 Sun, 21 Dec 2008
[akk5] A. Krzysztof Kwaśniewski, Some Cobweb Posets Digraphs' Elementary Properties and Questions ArXiv:0812.4319 Tue, 23 Dec 2008
[eks1] Ewa Krot-Sieniawska, Reduced Incidence algebras description of cobweb posets and KoDAGs, arXiv:0802.4293 Fri, 29 Feb 2008
[eks2] Ewa Krot-Sieniawska, On incidence algebras description of cobweb posets, arXiv:0802.3703 Tue, 26 Feb 2008
[eks3] Ewa Krot-Sieniawska, Characterization of Cobweb Posets as KoDAGs, arXiv:0802.2980 Thu, 21 Feb 2008
[eks4] Ewa Krot-Sieniawska, On Characteristic Polynomials of the Family of Cobweb Posets, arXiv:0802.2696 Tue, 19 Feb 2008
prof A. Krzysztof Kwaśniewski home page - home page of Cobweb Posets' concept's author