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Voting, The Symmetric Group, And Representation Theory, Zajj Daugherty '05, Alexander K. Eustis '06, Gregory Minton '08, Michael E. Orrison
Voting, The Symmetric Group, And Representation Theory, Zajj Daugherty '05, Alexander K. Eustis '06, Gregory Minton '08, Michael E. Orrison
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We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied QSn-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory.
Counting On Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
Counting On Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
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Chebyshev polynomials have several elegant combinatorial interpretations. Specificially, the Chebyshev polynomials of the first kind are defined by T0(x) = 1, T1(x) = x, and Tn(x) = 2x Tn-1(x) - Tn-2(x). Chebyshev polynomials of the second kind Un(x) are defined the same way, except U1(x) = 2x. Tn and Un are shown to count tilings of length n strips with squares and dominoes, where the tiles are given weights and sometimes color. Using these interpretations, many identities satisfied by Chebyshev polynomials can be given …