Monthly Archives: February 2011

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

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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

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correlations between Spearman’s rho p distances for various p: a plethora of central limit theorems

In the past week or so I have been playing around with various metrics on , as a continuation of the previous post. Most notably I looked at the Spearman’s distance, which is the analogue of the footrule and defined … Continue reading

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History of limit theorems for rank correlations and some tedious computations

Suppose we sample pairs of values from some bivariate distribution. We would like to understand the correlation between the two random variables represented by the first and second coordinates respectively. One way to do this is of course to compute … Continue reading

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Algebraic combinatorics Lecture 12 part 2: combinatorial Hopf algebra, examples

It is an attempt to algebraicize combinatorics. First we look at an example from Rota. Given a field with char . Let be the vector space over with a basis indexed by the set of all ranked posets. Definition 2 … Continue reading

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Algebraic combinatorics lecture 12 part 1: Convergence of riffle shuffle using quasi-symmetric function theory

Recall we have two bases for quasi-symmetric functions, where is a composition (i.e., the order of the entries actually matter) and is the subset of associated with . We proved last time that , where the left hand side is … Continue reading

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Bolthausen’s proof of Berry Esseen theorem

In this section we re-present Bolthausen’s original proof of Berry Esseen using Stein’s method and fill in some computational details. I hope to make the argument more transparent to people trying to learn Stein’s method and Berry-Esseen theorem. The letter … Continue reading

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Quasi-symmetric (QS) functions and generalized riffle shuffle

An example of a QS function is given by which is clearly not symmetric. In general, is quasi-symmetric if for all and : . Let be the space of QS functions of degree with infinitely many variables as usual. And … Continue reading

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Some linear algebra fact: Schur complement formula and projection operator

One of the most important thing is Schur complement formula: suppose we have a square -block matrix of arbitrary block size: then . The proof is very simple, and consists of factoring out from and then doing a block column … Continue reading

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Algebraic combinatorics Lecture 9: symmetric group representations and isomorphism of symmetric functions and class functions

Since I am not a superhuman, my blog lengths will significantly reduce in size from now on. Let be the space of functions on a finite group . We can define the usual inner product on it, and the convolution … Continue reading

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