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seminars:comb:abstract.200603sag

Let S_{n} be the symmetric group of all permutations of {1,2,…,n}. A permutation π = a_{1} a_{2} … a_{n} in S_{n} (written in one-line form) has major index

maj π = Sum_{a<sub>i} > a_{i+1}</sub> i,

i.e., maj π is the sum of all the indices i where π has a descent. The major index is an important statistic in combinatorics and has many interesting properties.

Now fix two positive integers k, l which are relatively prime (i.e., have no common factors) and are at most n. Let m_{n}^{k,l} be the k×l matrix whose (i,j) entry is the cardinality of the set

{π in S_{n} : maj π = i (mod k) and maj π^{-1} = j (mod l)}.

Surprisingly, this matrix has all its entries equal! We will outline a combinatorial proof of this theorem and other related results.

This is joint work with Helene Barcelo and Sheila Sundaram.

seminars/comb/abstract.200603sag.txt · Last modified: 2020/01/29 14:03 (external edit)

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