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Mathematics

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Combinatorics

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Nonsupereulerian Graphs With Large Size, Paul A. Catlin, Zhi-Hong Chen Oct 2019

Nonsupereulerian Graphs With Large Size, Paul A. Catlin, Zhi-Hong Chen

Zhi-Hong Chen

No abstract provided.

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Even Subgraphs Of A Graph, Hong-Jian Lai, Zhi-Hong Chen Oct 2019

Even Subgraphs Of A Graph, Hong-Jian Lai, Zhi-Hong Chen

Zhi-Hong Chen

No abstract provided.

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Properties Of Catlin’S Reduced Graphs And Supereulerian Graphs, Wei-Guo Chen, Zhi-Hong Chen, Mei Lu Sep 2019

Properties Of Catlin’S Reduced Graphs And Supereulerian Graphs, Wei-Guo Chen, Zhi-Hong Chen, Mei Lu

Zhi-Hong Chen

A graph G is called collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is the reduction of G if it is obtained from G by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs G of order n with d(u) + d(v) ≥ 2(n/p − …

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Notes On The Proof Of The Van Der Waerden Permanent Conjecture, Vicente Valle Martinez Apr 2018

Notes On The Proof Of The Van Der Waerden Permanent Conjecture, Vicente Valle Martinez

Vicente Valle Martinez

The permanent of an $n\times n$ matrix $A=(a_{i j})$ with real entries is defined by the sum
$$\sum_{\sigma \in S_n} \prod_{i=1}^{n} a_{i \sigma(i)}$$
where $S_n$ denotes the symmetric group on the $n$-element set $\{1,2,\dots,n\}$.
In this creative component we survey some known properties of permanents, calculation of permanents for particular types of matrices and their applications in combinatorics and linear algebra. Then we follow the lines of van Lint's exposition of Egorychev's proof for the van der Waerden's conjecture on the permanents of doubly stochastic matrices. The purpose of this component is to provide elementary proofs of several interesting known …

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An Alternate Approach To Alternating Sums: A Method To Die For, Arthur T. Benjamin, Jennifer J. Quinn Nov 2017

An Alternate Approach To Alternating Sums: A Method To Die For, Arthur T. Benjamin, Jennifer J. Quinn

Jennifer J. Quinn

No abstract provided in this article.

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Fibonacci Deteminants - A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn Nov 2017

Fibonacci Deteminants - A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn

Jennifer J. Quinn

In this paper, we provide combinatorial interpretations for some determinantal identities involving Fibonacci numbers. We use the method due to Lindström-Gessel-Viennot in which we count nonintersecting n-routes in carefully chosen digraphs in order to gain insight into the nature of some well-known determinantal identities while allowing room to generalize and discover new ones.

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Unevening The Odds Of "Even Up", Arthur T. Benjamin, Jennifer J. Quinn Feb 2014

Unevening The Odds Of "Even Up", Arthur T. Benjamin, Jennifer J. Quinn

Jennifer J. Quinn

No abstract provided in this article.

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Paint It Black -- A Combinatorial Yawp, Arthur T. Benjamin, Jennifer J. Quinn, James A. Sellers, Mark A. Shattuck Feb 2014

Paint It Black -- A Combinatorial Yawp, Arthur T. Benjamin, Jennifer J. Quinn, James A. Sellers, Mark A. Shattuck

Jennifer J. Quinn

No abstract provided in this paper.

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A Stirling Encounter With Harmonic Numbers, Arthur T. Benjamin, Gregory O. Preston '01, Jennifer J. Quinn Feb 2014

A Stirling Encounter With Harmonic Numbers, Arthur T. Benjamin, Gregory O. Preston '01, Jennifer J. Quinn

Jennifer J. Quinn

No abstract provided in this article.

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Counting On Continued Fractions, Arthur T. Benjamin, Francis E. Su, Jennifer J. Quinn Feb 2014

Counting On Continued Fractions, Arthur T. Benjamin, Francis E. Su, Jennifer J. Quinn

Jennifer J. Quinn

No abstract provided in this article.

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Phased Tilings And Generalized Fibonacci Identities, Arthur T. Benjamin, Jennifer J. Quinn, Francis E. Su Feb 2014

Phased Tilings And Generalized Fibonacci Identities, Arthur T. Benjamin, Jennifer J. Quinn, Francis E. Su

Jennifer J. Quinn

Fibonacci numbers arise in the solution of many combinatorial problems. They count the number of binary sequences with no consecutive zeros, the number of sequences of 1's and 2's which sum to a given number, and the number of independent sets of a path graph. Similar interpretations exist for Lucas numbers. Using these interpretations, it is possible to provide combinatorial proofs that shed light on many interesting Fibonacci and Lucas identities (see [1], [3]). In this paper we extend the combinatorial approach to understand relationships among generalized Fibonacci numbers. Given G0 and G1 a generalized Fibonacci sequence G0, G1, G2,... …

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Summing Cubes By Counting Rectangles, Arthur T. Benjamin, Jennifer J. Quinn, Calyssa Wurtz Feb 2014

Summing Cubes By Counting Rectangles, Arthur T. Benjamin, Jennifer J. Quinn, Calyssa Wurtz

Jennifer J. Quinn

No abstract provided in this article.

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The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn Feb 2014

The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn

Jennifer J. Quinn

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.

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