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Physical Sciences and Mathematics Commons™
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Articles 1 - 4 of 4
Full-Text Articles in Physical Sciences and Mathematics
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
All HMC Faculty Publications and Research
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.
Almost Avoiding Permutations, Robert Brignall, Shalosh B. Ekhad, Rebecca Smith, Vincent Vatter
Almost Avoiding Permutations, Robert Brignall, Shalosh B. Ekhad, Rebecca Smith, Vincent Vatter
Dartmouth Scholarship
The permutation π of length n, written in one-line notation as π (1)π (2)· · · π (n), is said to contain the permutation σ if π has a subsequence that is order isomorphic to σ, and each such subsequence is said to be an occurrence of σ in π or simply a σ pattern. For example, π = 491867532 contains σ = 51342 because of the subsequence π (2)π (3)π (5)π (6)π (9) = 91672. Permutation containment is easily seen to be a partial order on the set of all (finite) permutations, which we simply denote by ≤. If …
Counting On Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
Counting On Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
All HMC Faculty Publications and Research
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 …
Hopf Quivers And Nichols Algebras In Positive Characteristic, Claude Cibils, Aaron Lauve, Sarah Witherspoon
Hopf Quivers And Nichols Algebras In Positive Characteristic, Claude Cibils, Aaron Lauve, Sarah Witherspoon
Mathematics and Statistics: Faculty Publications and Other Works
We apply a combinatorial formula of the first author and Rosso, for products in Hopf quiver algebras, to determine the structure of Nichols algebras. We illustrate this technique by explicitly constructing new examples of Nichols algebras in positive characteristic. We further describe the corresponding Radford biproducts and some liftings of these biproducts, which are new finite dimensional pointed Hopf algebras.