Category Archives: algebraic combinatorics

based on Persi Diaconis’ lectures

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

Posted in algebraic combinatorics | Tagged , | Leave a comment

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 | Leave a comment

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

Posted in algebraic combinatorics, Uncategorized | Leave a comment

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 | 5 Comments

Algebraic combinatorics Lecture 12: Dehn-Sommerville, flag enumeration and Eulerian posets; the tip of a convex iceberg

Let Q be a convex polytope in , and let be the number of i-dimensional 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

Posted in algebraic combinatorics, Uncategorized | Leave a comment

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 , , | 3 Comments

Algebraic combinatorics lecture 13: Quasi-symmetric 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

Posted in algebraic combinatorics, Uncategorized | Leave a comment