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Category Archives: algebraic combinatorics
Cauchy’s Series product identity, an exercise from Macdonald’s book on symmetric functions
After seeing how diminished my daily visitor count has gone, since the last series of rapid fire posting of technical posts on CS related issues, I decide to come back to mathematics. The result is motivated by an example/exercise from … Continue reading
Greene’s theorem, Knuth moves, and Jeu de Taquin
Lemma 6 do not change under Knuth moves. Proof: and are related by , where if , then . This follows fromm the definition of .Therefore if and only if . Fix , suppose the lemma is false. Say . … Continue reading
Posted in algebraic combinatorics, Uncategorized
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Polya theory continued
The setting is as previous lecture. Many of the expositions are adapted from Stanley’s Volume 2 appendix by Sergey Fomin. , where and denotes the size of the inverse image by an abuse of notation. Last time we saw that … Continue reading
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Algebraic combinatorics lecture 14: Polya Theory
We are given a domain , a finite set , and a group acting on , acting on . Let and , i.e., acts on , and splits it into orbits. Visually it’s easiest to represent as coloring schemes of … Continue reading
Posted in algebraic combinatorics, Uncategorized
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Algebraic combinatorics Lecture 12: DehnSommerville, flag enumeration and Eulerian posets; the tip of a convex iceberg
Let Q be a convex polytope in , and let be the number of idimensional faces. How are the related? Of course Euler’s relation states and for convex polytope, the genus is . Thus for , we can parametrize all … Continue reading
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Plane partitions, Lozenge tiling, and MacMahon’s formula
Recall a partition is given by a sequence of weakly decreasing, positive integers . A plane partition, on the other hand, is given by a Young tableau like object where each row and each column is weakly decreasing, with repeated … Continue reading
Posted in algebraic combinatorics, Uncategorized
Tagged combinatorics, Diaconis, symmetric functions
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Algebraic combinatorics lecture 13: Quasisymmetric functions as a Hopf algebra, generalized birthday problem
Recall the notion of dual of a finite dimensional vector space. Theorem 4 If C is a cogebra, with , then the vector sapce dual is an algebra with product and unit , defined by The unit is defined in … Continue reading
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