A note on V-binomials recurrence for Lucas companion to $U_n$ sequence $V_n$
The looked for V-binomials recurrence for Lucas companion to $U_n$ sequence $V_n$ is delivered
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arXiv subjects
A. K. Kwasniewski
Publications and source records attributed to A. K. Kwasniewski.
On compositions of numbers and graphs
The main purpose of this note is to pose a couple of problems which are easily formulated thought some seem to be not yet solved. These problems are of general interest for discrete mathematics including a new twig of a bough of theory of graphs i.e. a given graph compositions. The problems result from and are served in the entourage of series of exercises with hints based predominantly on the second reference and other related recent papers.
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Graded posets inverse zeta matrix formula
We derive the explicit formula for the inverse of zeta matrix for any graded posets with the finite set of minimal elements . The combinatorial interpretation of this result is given. For that to do special number theoretic code triangles for graded posets are proposed and apart from the present author combinatorial interpretation of $F-nomial$ coefficients another one is proposed referring to the number of all maximal chains in the corresponding poset intervals.
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Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices
Natural join of $di-bigraphs$ that is directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their $Hasse$ digraphs called $KoDAGs$. $KoDAGs$ are special orderable directed acyclic graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. $KoDAGs$ appear to be distinguished family of $Ferrers$ digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called $di-bicliques$. These digraphs serve to represent faithfully corresponding relations of arbitrary arity so that all relations of arbitrary arity are their subrelations. Being this $chain -way$ complete if compared with kompletne $Kuratowski$ bipartite graphs their $DAG$ denotation is accompanied with the letter $K$ in front of descriptive abbreviation $oDAG$. The way to join bipartite digraphs of binary into $multi-ary$ relations is the natural join operation either on relations or their digraph representatives. This natural join operation is denoted here by $\os$ symbol deliberately referring to the direct sum $\oplus$ of adjacency matrices as it becomes the case for disjoint $di-bigraphs$.
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Natural join construction of graded posets versus ordinal sum and discrete hyper boxes
One introduces here the natural join $P \os Q$ of graded posets $< P,\leq_P >$ and $< Q,\leq_Q >$ with correspondingly maximal and minimal sets being identical as expressed by ordinal sum $P\oplus Q$ apart from other definition and due to that one arrives at a simple proof of the $M{ö}bius $ function formula for cobweb posets. We also quote the other authors explicit formulas for the zeta matrix and its inverse for any graded posets with the finite set of minimal elements from earlier works of the author. These formulas are based on the formulas for cobweb posets and their $Hasse$ diagrams or graphs named $KoDAGs$ which are interpreted as chains of binary complete or universal relations joined by the natural join operation. Natural join of two independent sets is therefore the ordinal sum of this trivially ordered posets represented also by directed biclique named dibiclique and correspondingly by their $Hasse $ diagrams or graphs named $KoDAGs$. Such cobweb posets and equivalently their Hasse diagrams or graphs named $KoDAGs$ are also encoded by discrete hyper-boxes and the natural join operation of such discrete hyper boxes is just cartesian product of them accompanied with projection out of common faces. All graded posets with no mute vertices in their $Hasse$ diagrams which means that no vertex has indegree or outdegree equal zero are natural join of chain of relations and may be at the same time interpreted an $n-ary$ relation, $n \in N \cup \{\infty \}$.
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Graded posets zeta matrix formula
The way to arrive at formula of zeta matrix for any graded posets with the finite set of minimal elements is delivered following the first reference. This is being achieved via adjacency and zeta matrix description of bipartite digraphs chains, the representatives of graded posets. The bipartite digraphs elements of such chains amalgamate to form corresponding cover relation graded poset digraphs with corresponding adjacency matrices being amalgamated throughout natural join as special adequate database operation. The colligation of reachability and connectivity with the presented description is made explicit. The special posets encoded via kodags directed acyclic graphs as cobeb posets cover relations digraphs are recognized as an example of differential posets subfamily. As on this night one reminisce anniversary of death of distinguished johann bernoulli the first this sylvester night article is to commemorate this date.
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On cobweb posets most relevant codings
One considers here orderable acyclic digraphs named KoDAGs which represent the outmost general chains of dibicliques denoting thus the outmost general chains of binary relations. Because of this fact KoDAGs start to become an outstanding concept of nowadays investigation. We propose here examples of codings of KoDAGs looked upon as infinite hyper-boxes as well as chains of rectangular hyper-boxes in N^\infty. Neither of KoDAGs codings considered here is a poset isomorphism with Pi = . Nevertheless every example of coding supplies a new view on possible investigation of KoDAGs properties. The codes proposed here down are by now recognized as most relevant codes for practical purposes including visualization. More than that. Employing quite arbitrary sequences F=\{n_F\}_{n\geq 0} infinitely many new representations of natural numbers called base of F number system representations are introduced. These constitute mixed radix-type numeral systems. F base nonstandard positional numeral systems in which the numerical base varies from position to position have picturesque interpretation due to KoDAGs graphs and their correspondent posets which in turn are endowed on their own with combinatorial interpretation of uniquely assigned to KoDAGs F-nomial coefficients. The base of F number systems are used for KoDAGs coding and are interpreted as chain coordinatization in KoDAGs pictures as well as systems of infinite number of boxes sequences of F-varying containers capacity of subsequent boxes. Needless to say how crucial is this base of F number system for KoDAGs hence consequently for arbitrary chains of binary relations. New F based numeral systems are umbral base of F number systems in a sense to be explained in what follows.
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Fibonomial cumulative connection constants
In this note we present examples of cumulative connection constants included new fibonomial ones. All examples posses combinatorial interpretation.
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How the work of Gian Carlo Rota had influenced my group research and life
One outlines here in brief how the work of $Gian Carlo Rota$ had influenced my group research and life, starting from the end of the last century up to present time state of $The Internet Gian Carlo Rota Polish Seminar$. This note has been written for the $Rota Memorial Conference$ to be held on 16-18 Feb 2009, Milan, Italy.
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New formulas for Stirling-like numbers and Dobinski-like formulas
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$. These extensions naturally encompass the well known $q$- extensions. The indicatory references are to point at a part of the vast domain of the foundations of computer science in arxiv affiliation.
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On cobweb posets and their combinatorially admissible sequences
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.
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Extended finite operator calculus as an example of algebraization of analysis
A calculus of sequences started by professor morgan ward constitutes the general scheme for extensions of classical operator calculus of the distinguished gian carlo rota considered by many afterwards and after ward morgan. Because of the historically now established notation we call the wardian calculus of sequences in its afterwards elaborated form a psi calculus. The psi calculus in parts appears to be almost automatic, natural extension of classical operator calculus or equivalently of umbral calculus . This is a review article based on the turn of the centuries author relevant contributions.
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More on Algebraic Structure of the Complete Partition Function for the $ Z_n $ - Potts Model, Part 1
In this first part of a larger review undertaking the results of the first author and a part of the second author doctor dissertation are presented. Next we plan to give a survey of a nowadays situation in the area of investigation. Here we report on what follows. Calculation of the partition function for any vector potts model is at first reduced to the calculation of traces of products of the generalized clifford algebra generators. The formula for such traces is derived. This enables one, in principle, to use an explicit calculation algorithm for partition functions also in other models for which the transfer matrix is an element from generalized clifford algebra. The method - simple for $Z_2$ case - becomes complicated for $Z_n$, $n>2$, however everything is controlled, in principle, due to knowledge of the corresponding algebra properties and those of generalized cosh function. The discussion of the content of the in statu nascendi second part is to be found at the end of this presentation. This constitutes the last section.
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Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics
Before we dive into the accessibility stream of nowadays indicatory applications of octonions to computer and other sciences and to quantum physics let us focus for a while on the crucially relevant events for todays revival on interest to nonassociativity. Our reflections keep wandering back to the $Brahmagupta$ $Fibonacc$ two square identity and then via the $Euler$ four square identity up to the $Degen$ $Ggraves$ $Cayley$ eight square identity. These glimpses of history incline and invite us to retell the story on how about one month after quaternions have been carved on the $Broughamian$ bridge octonions were discovered by $John$ $Thomas$ $Ggraves$, jurist and mathematician, a friend of $William$ $Rowan$ $Hamilton$. As for today we just mention en passant quaternionic and octonionic quantum mechanics, generalization of $Cauchy$ $Riemann$ equations for octonions and triality principle and $G_2$ group in spinor language in a descriptive way in order not to daunt non specialists. Relation to finite geometries is recalled and the links to the 7stones of seven sphere, seven imaginary octonions units in out of the $Plato$ cave reality applications are appointed . This way we are welcomed back to primary ideas of $Heisenberg$, $Wheeler$ and other distinguished fathers of quantum mechanics and quantum gravity foundations.
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Cauchy_Riemann Equations for Cayley Numbers` Functions
Since the discovery of octonions in 1843 we seem to be still lacking a satisfactory if any theory of octave valued functions satisfactory according to standard requirements or expectation from the side of a theory like a one might look for. Here is a proposal coming back to my twentieth century presentation of a perhaps nonstandard idea hoping to be coping with nonassociativity by an invention.
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Comments on combinatorial interpretation of fibonomial coefficients - an email style letter
Up to our knowledge -since about 126 years we were lacking of classical type combinatorial interpretation of Fibonomial coefficients as it was Lukas \cite{1} - to our knowledge -who was the first who had defined Finonomial coefficients and derived a recurrence for them (see Historical Note in \cite{2,3}). Here we inform that a join combinatorial interpretation was found \cite{4} for all binomial-type coefficient - Fibonomial coefficients included.
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New type Stirling like numbers - an email style letter
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 extension so as to encompass the new type combinatorial objects from [2] as leading examples. Recently the present author had introduced also [6] two natural partial orders in there: one $\leq$ in grading-natural subsets of cobweb`s prefabs sets [2] and in the second proposal one endows the set sums of the so called "prefabiants" with such another partial order that one arrives at Bell-like numbers including Fibonacci triad sequences introduced by the present author in [7]. Here we quote the basic observations concerning the new type Stirling like numbers as they appear in [6]. For more on notation, Stirling like numbers of the first kind and for proofs - see [6].
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Ivan Bernoulli Series Universalissima
The taylor formula pertains historically also to johann or ivan bernoulli. Bernoulli series uiversalisima appeared in acta eruditorum in leipzig when brook taylor was nine years old. As on today one affirms two hundred fifty eight anniversary of death of johann bernoulli this article is proposed also pro memoriam. Very recent extensions of this celebrated formula are indicated in the framework of extended umbral calculus.
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