Combinatorial principles
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On the sum of parts with multiplicity at least 2 in all the partitions of n Open
In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is…
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Combinatorial Flows and Their Normalisation Open
This paper introduces combinatorial flows that generalize combinatorial proofs such that they also include cut and substitution as methods of proof compression. We show a normalization procedure for combinatorial flows, and how syntactic p…
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Combinatorial properties of poly-Bernoulli relatives Open
In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…
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Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application Open
In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic gener…
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Combinatorial Properties of Poly-Bernoulli Relatives Open
See the abstract in the attached pdf.
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THE DEFINABILITY STRENGTH OF COMBINATORIAL PRINCIPLES Open
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a defin…
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Combinatorial Interpretation of Numbers in the Generalized Padovan Sequence and Some of Its Extensions Open
There is ongoing research into combinatorial methods and approaches for linear and recurrent sequences. Using the notion of a board defined for the Fibonacci sequence, this work introduces the Padovan sequence combinatorial approach. Thus,…
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Combinatorial Proofs of Addition Formulas Open
In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted pa…
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A Combinatorial Proof of a Schmidt Type Theorem of Andrews and Paule Open
This note is devoted to a combinatorial proof of a Schmidt type theorem due to Andrews and Paule. A four-variable refinement of Andrews and Paule's theorem is also obtained based on this combinatorial construction.
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Combinatorial Exploration: An algorithmic framework for enumeration Open
Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works an…
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A New Approach to the $r$-Whitney Numbers by Using Combinatorial Differential Calculus Open
In the present article we introduce two new combinatorial interpretations of the $r$-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar $G:=\{ y\rightarrow yx^{m}, x\r…
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On a generalized basic series and Rogers-Ramanujan type identities Open
In this paper, we give the generalization of MacMahon's type combinatorial identities. A generalized $q$-series is interpreted as the generating function of two different combinatorial objects, viz., restricted $n$-color partitions and wei…
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Extensions of the combinatorics of poly-Bernoulli numbers Open
In this paper we extend the combinatorial theory of poly-Bernoulli numbers. We prove combinatorially some identities involving poly-Bernoulli polynomials. We introduce a combinatorial model for poly-Euler numbers and provide combinatorial …
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Using Periodicity Properties to Generate the Combinatorial Configurations Open
Identifying patterns of the ordering of a certain combinatorial set allows to develop of simple procedures for its generation for an arbitrary value and to strictly prove that this set contains all non-identical combinatorial configuration…
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A bialgebraic characterization of symmetric powers in $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal categories Open
In any symmetric monoidal category, the $n$-th (co)equalizer symmetric power of an object $A$ is the (co)equalizer of all the permutations from $A^{\otimes n}$ to itself. If the symmetric monoidal category is $\mathbb{Q}_{\ge 0}$-linear, t…
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Some combinatorial matrices and their LU-decomposition Open
Three combinatorial matrices were considered and their LU-decompositions were found. This is typically done by (creative) guessing, and the proofs are more or less routine calculations.
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Computational Complexity in Algebraic Combinatorics Open
Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives and brought us some …
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Depth lower bounds in Stabbing Planes for combinatorial principles Open
Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so …
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Combinatorial interpretations of two identities of Guo and Yang Open
The restricted partitions in which the largest part is less than or equal to $N$ and the number of parts is less than or equal to $k$ were investigated by Andrews in \cite{Andrews76}. These partitions were extended recently by the author t…
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A Note on Generalization of Combinatorial Identities Due to Gould and Touchard Open
Using a hypergeometric series approach, a general combinatorial identity is found in this note, and among its special cases are well-known and classical combinatorial identities due to Gould and Touchard.
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A NOTE OF THE COMBINATORIAL INTERPRETATION OF THE PERRIN AND TETRARRIN SEQUENCE Open
The present study carries out an investigation around the Perrin and Tetrarrin numbers, allowing a combinatorial interpretation for these sequences. Furthermore, it is possible to establish a study around the respective polynomial numbers …
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Language Combinatorics: Aspects of Studying Open
The article is devoted to the identification and characterization of aspects of the language combinatorics studying. The research is conducted in the framework of combinatorial linguistics that studies the linear relations of language unit…
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Characterizing large cardinals through Neeman's pure side condition forcing Open
We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $ω_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combin…
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From the History of Combinatorial Analysis: From Idea to Research Schools Open
The article explores the development of combinatorial analysis from the idea to scientific schools. Combinatorial research was stimulated by G.W. Leibniz's ideas about combinatorial art and special geometric analysis – Analysis Situs in th…
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Well ordering principles for iterated $Π^1_1$-comprehension Open
We introduce ordinal collapsing principles that are inspired by proof theory but have a set theoretic flavor. These principles are shown to be equivalent to iterated $Π^1_1$-comprehension and the existence of admissible sets, over weak bas…
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Combinatorial Proofs of Identities of Alzer and Prodinger and Some Generalizations Open
See the abstract in the attached pdf.
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Combinatorial interpretations of two identities of Guo and Yang Open
The restricted partitions in which the largest part is less than or equal to $N$ and the number of parts is less than or equal to $k$ were investigated by Andrews. These partitions were extended recently by the author to partitions into pa…
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Reductions of well-ordering principles to combinatorial theorems Open
A well-ordering principle is a principle of the form: If $X$ is well-ordered then $F(X)$ is well-ordered, where $F$ is some natural operator transforming linear orders into linear orders. Many important subsystems of Second-order Arithmeti…
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Full runner removal theorem for Ariki-Koike algebras Open
We consider the representation theory of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group Cr≀Sn. We define the addition of a runner full of beads for the abacus display of a multipartition and i…
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Two Formulas for $F$-Polynomials Open
We discuss a product formula for $F$-polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock-Goncharov decomposition of mutations. …