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Full-Text Articles in Mathematics
Recounting Determinants For A Class Of Hessenberg Matrices, Arthur T. Benjamin, Mark A. Shattuck
Recounting Determinants For A Class Of Hessenberg Matrices, Arthur T. Benjamin, Mark A. Shattuck
All HMC Faculty Publications and Research
We provide combinatorial interpretations for determinants which are Fibonacci numbers of several recently introduced Hessenberg matrices. Our arguments make use of the basic definition of the determinant as a signed sum over the symmetric group.
A Combinatorial Solution To Intertwined Recurrences, Arthur T. Benjamin, Michael D. Hirschhorn
A Combinatorial Solution To Intertwined Recurrences, Arthur T. Benjamin, Michael D. Hirschhorn
All HMC Faculty Publications and Research
We provide combinatorial derivations of solutions to intertwined second order linear recurrences (such as an = pbn-1 + qan-2, bn = ran-1 + sbn-2) by counting tilings of length n strips with squares and dominoes of various colors and shades. A similar approach can be applied to intertwined third order recurrences with coefficients equal to one. Here we find that all solutions can be expressed in terms of tribonacci numbers. The method can also be easily extended to solve and combinatorially comprehend kth order Fibonacci recurrences.
Fibonacci Deteminants - A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn
Fibonacci Deteminants - A Combinatorial Approach, Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn
All HMC Faculty Publications and Research
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.