Arthur T. Benjamin, Doron Zeilberger
In this note, we prove that every prime of the form 4m + 1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued...
LINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS (2009)
Abstract. We present a tiling interpretation for k th order linear recurrences, which yields new combinatorial proofs for recurrence identities. Moreover, viewing the tiling process as a Markov chain...
Arthur T. Benjamin, Jennifer J. Quinn, Francis Edward Su
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...
182 MATHEMATICS MAGAZINE The Fibonacci Numbers— Exposed More Discretely (2008)
Arthur T. Benjamin, Jennifer J. Quinn, Fm Fgcd(n
In the previous article, Kalman and Mena [5] propose that Fibonacci and Lucas sequences, despite the mathematical favoritism shown them for their abundant patterns, are nothing more than ordinary...
A Probabilistic View of Certain Weighted Fibonacci Sums (2008)
Arthur T. Benjamin, Judson D. Neer, Daniel E. Otero, James A. Sellers
In this paper we investigate sums of the form an: = � k≥1
Arthur T. Benjamin, Doron Zeilberger
In this note, we prove that every prime of the form 4m + 1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued...
Combinatorial interpretations of spanning tree identities (2008)
Arthur T. Benjamin, Carl R. Yerger
We present a combinatorial proof that the wheel graph Wn has L2n − 2 spanning trees, where Ln is the nth Lucas number, and that the number of spanning trees of a related graph is a Fibonacci...
CATALAN DETERMINANTS — A COMBINATORIAL APPROACH (2008)
Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn, Carl R
Abstract. Determinants of matrices involving the Catalan sequence have appeared throughout the literature. In this paper, we focus on the evaluation of Hankel determinants featuring Catalan numbers...
1. THE PROBLEM OF THE DETERMINED ANTS. Imagine four determined (2008)
Arthur T. Benjamin, Naiomi T. Cameron
ants who simultaneously walk along the edges of the picnic table graph of Figure 1. The ants can move only to the right (northeast, southeast, and sometimes due east) with the goal of reaching four...
A combinatorial approach to hyperharmonic numbers (2008)
Arthur T. Benjamin, David Gaebler, Robert Gaebler
Hyperharmonic numbers arise by taking repeated partial sums of harmonic numbers. These numbers can be expressed in terms of r-Stirling numbers, leading to combinatorial interpretations of many...
A combinatorial approach to hyperharmonic numbers (2008)
Arthur T. Benjamin, David Gaebler, Robert Gaebler
Hyperharmonic numbers arise by taking repeated partial sums of harmonic numbers. These numbers can be expressed in terms of r-Stirling numbers, leading to combinatorial interpretations of many...
e-mail address: jquinnQoxy. edu (2008)
Jennifer J. Quinn, Arthur T. Benjamin
e-mail address: benjaminQhmc.edu The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 5 k 5 1 5 m,...
SELF-AVOIDING WALKS AND FIBONACCI NUMBERS (2008)
Abstract. By combinatorial arguments, we prove that the number of selfavoiding walks on the strip {0, 1} × Z is 8Fn − 4 when n is odd and is 8Fn − n when n is even. Also, when backwards moves...
196 MATHEMATICS MAGAZINE The Probability of Relatively Prime Polynomials (2008)
Euclid Does Integers, Arthur T. Benjamin, Curtis D. Bennett
The Euclidean algorithm for finding greatest common divisors, one of the oldest algorithms in the world, is also one of the most versatile. When applied to integers, Euclid’s theorem can be stated...
Optimal Token Allocations in Solitaire Knock ’m Down (2008)
Arthur T. Benjamin, Matthew T. Fluet, Mark L. Huber
In the game Knock ’m Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is...
LINEAR RECURRENCES THROUGH TILINGS AND MARKOV CHAINS (2007)
Abstract. We present a tiling interpretation for k th order linear recurrences, which yields new combinatorial proofs for recurrence identities. Moreover, viewing the tiling process as a Markov chain...
Pythagorean primes and palindromic continued fractions, Integers 5 (2005)
Arthur T. Benjamin, Doron Zeilberger
In this note, we prove that every prime of the form 4m+1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued...
Transformation of Statistics in Fractional Quantum Hall Systems (2000)
Quinn, John J., Wojs, Arkadiusz, Quinn, Jennifer J., Benjamin, Arthur T.
A Fermion to Boson transformation is accomplished by attaching to each Fermion a tube carrying a single quantum of flux oriented opposite to the applied magnetic field. When the mean field...
Counting On Continued Fractions (2000)
Arthur T. Benjamin, Jennifer J. Quinn, Francis Edward Su, Edward Su
this paper, we provide a combinatorial interpretation for the numerators and denominators of continued fractions which makes this reversal phenomenon easy to see. Our interpretation also allows us to...
Phased Tilings and Generalized Fibonacci Identities (2000)
Arthur T. Benjamin, Jennifer J. Quinn, Francis Edward Su, Edward Su
We interpret generalized Fibonacci numbers as phased tilings and introduce several combinatorial techniques which provide new proofs for a host of identities. These follow naturally as the phased...
NOTE Composite Fermions and Integer Partitions (2000)
Arthur T. Benjamin, Jennifer J. Quinn, John J. Quinn, Arkadiusz Wjs
We utilize the KOH theorem to prove the unimodality of integer partitions with at most a parts, all parts less than or equal to b, that are required to contain either repeated or consecutive parts....
Counting on continued fractions (2000)
Arthur T. Benjamin, Francis Edward Su, Jennifer J. Quinn
You might be surprised to learn that the finite continued fraction 3 + 7 +
Matthew T. Fluet, Advisor Prof, Arthur T. Benjamin
Knock ’m Down is a game of dice that is so easy to learn that it is being played in classrooms around the world as a way to develop students ’ intuition about probability. However, as analysis...