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Monthly Archives: January 2011
Algebraic combinatorics lecture 6: RS(K), Dual RSK, and analytic definition of Schur functions
First we have an easy result Proposition 1 where is the number of SSYT of shape and type . Proof: We know that Equating coefficient of both sides, we get on LHS and on RHS. Thus we also have . … Continue reading
Posted in algebraic combinatorics, Uncategorized
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Algebraic combinatorics lecture 5: RSK algorithm
First we describe the Schensted algorithm, which associates to each permutation two Young tableaux of the same shape but possibly different types; it consists of bumping numbers into a greedy patience sorting procedure. In the first step we write down. … Continue reading
Posted in algebraic combinatorics, Uncategorized
Tagged combinatorics, Diaconis, RSK algorithm, symmetric functions
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Algebraic combinatorics Lecture 4: patience sorting, symmetry of schur functions and Kostka numbers
Recall the definition of skew Young tableaux and their Schur functions and . When , we get the usual Schur functions .The reason we consider skew tableaux is they are related to skew characters of to be discussed later. the … Continue reading
Posted in algebraic combinatorics, Uncategorized
Tagged card shuffling, combinatorics, Diaconis
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Graphical inference model: lecture 4. Max product algorithm and group testing
Max product is algorithm is closely related to the sum product algorithm, in that it uses partial marginals to compute the total marginal. But the objective is to find the mode of a distribution, i.e., . It does that by … Continue reading
Gurvits’ inequality and proof of Van der Waerden’s conjecture
Today I went to Don Knuth’s talk on some recent spectacular progress made in the problem of computing permanents. The talk is motivated and mainly based on a recent article in the American Math Monthly called “On Leonid Gurvits’s proof … Continue reading
Algebraic combinatorics Lecture 3
First it is useful to have the Hall inner product on , the space of all symmetric functions in infinitely many variables. It is defined by To show it’s symmetric, we check it on the basis . Then it follows … Continue reading
Posted in algebraic combinatorics, Uncategorized
Tagged combinatorics, Diaconis, symmetric functions
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Graphical inference model lecture 2
The indexing and bookkeeping of notations in this class have always been a nightmare to me. I try my best to state things correctly. Whenever possible I copy directly from Andrea’s notes. The moral of today’s lesson is that trees … Continue reading