Eigenvalues and Eigenvectors of the Matrix of Permutation Counts

2013/09/16 by Pawan Auorora, Auorora, Pawan, Shashank K Mehta +1
Mathematics · #15B99 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:15B99

paper · pdf · doi:10.48550/arxiv.1309.3836

Two figures

arxiv created 2013/09/20 · arxiv updated 2013/09/23

Abstract

Define a (n⁴+n²)/2× (n⁴+n²)/2 symmetric B. (ij)(kl) is an index where i,j,k,l∈ [n], (ab) is an unordered pair and (kl) is an ordered pair when i≠ j, otherwise it is also an unordered pair. B((ij)(kl),(ab)(xy)) is equal to the number of permutations of S_n in which min\i,j\ maps to k, max\i,j\ maps to l, min\a,b\ maps to x and max\a,b\ maps to y. We will show that B has four distinct eigenvalues: (3/2)n!, n(n-3)!, (n-1)!/(n-3), 2n(n-2)! and the corresponding eigenspace dimensions are 1, n-1\choose2², (n-1\choose2-1)², (n-1)² respectively.

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