Related papers: First Observations on Prefab Posets Whitney Number…
We introduce a natural partial order in structurally natural finite subsets the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like…
The notion of the Fibonacci cobweb poset from [1] has been naturally extended to any admissible sequence $F$ in [2] where it was also recognized that the celebrated prefab notion of Bender and Goldman [3] - (see also [4,5]) - admits such an…
In this paper we introduce a family of partitions of the set of natural numbers, Fibonacci-like partitions. In particular, we introduce a Fibonacci-like partition in a number of parts corresponding to the Fibonacci numbers, the standard…
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'…
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…
A class of new type graded infinite posets with minimal element are considered. These so called cobweb posets introduced recently by the present author provide a wide range of new noncommutative prefab combinatorial schema with…
Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$, we introduce the Fibonacci partition as a Fibonacci permutation of its blocks. Then we define…
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…
A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a…
In this paper, we study partitions of positive integers into distinct quasifibonacci numbers. A digraph and poset structure is constructed on the set of such partitions. Furthermore, we discuss the symmetric and recursive relations between…
Fibonomial coefficients count the number of specific finite birth self-similar subposets of an infinite non-tree poset naturally related to the Fibonacci tree of rabbits growth process.
The main purpose of this article is to pose three problems which are easy to be formulated in an elementary way. These problems which are specifically important also for the new class of partially ordered sets seem to be not yet solved.
We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers. Calculating sums of principal minors of the fixed order of the first type leads us to convolved Fibonacci numbers. Some identities for these and for…
We define combinatorially a partial order on the set partitions and show that it is equivalent to the Bruhat-Chevalley-Renner order on the upper triangular matrices. By considering subposets consisting of set partitions with a fixed number…
Primary definitions, notation and general observations of finite fibonomial operator calculus (ffoc) are presented. Kwasniewski's combinatorial interpretation of fibonomial coefficients by the use of fibonacci cobweb poset is given. Some…
A semiorder is a partially ordered set $P$ with two certain forbidden induced subposets. This paper establishes a bijection between $n$-element semiorders of length $H$ and $(n+1)$-node ordered trees of height $H+1$. This bijection…
The set $\Mfib$ of fibbinary numbers is defined via a bijection between the set $\BB{N}$ of natural numbers and $\Mfib$. Since the elements of $\Mfib$ do not exhaust $\BB{N}$, the structure of the complement $\overline{\Mfib}$ of $\Mfib$ in…
In recent Kwasniewski's papers inspired by O. V. Viskov it was shown that the $\psi$-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota - Mullin or equivalently - of umbral calculus of…
The explicite formulas for m\"{o}biusien function and some other important elements of the incidence algebra are delivered. For that to do one uses kwa\'sniewski's construction of his fibonacci cobweb poset in the plane grid coordinate…
In two previous papers we have presented partition formulae for the Fibonacci numbers motivated by the appearance of the Fibonacci numbers in the representation theory of the 3-Kronecker quiver and its universal cover, the 3-regular tree.…