Identifier
Values
=>
Cc0007;cc-rep
[]=>0
[[1]]=>1
[[1,2]]=>2
[[1],[2]]=>0
[[1,2,3]]=>3
[[1,3],[2]]=>0
[[1,2],[3]]=>1
[[1],[2],[3]]=>1
[[1,2,3,4]]=>4
[[1,3,4],[2]]=>0
[[1,2,4],[3]]=>1
[[1,2,3],[4]]=>2
[[1,3],[2,4]]=>0
[[1,2],[3,4]]=>1
[[1,4],[2],[3]]=>1
[[1,3],[2],[4]]=>0
[[1,2],[3],[4]]=>2
[[1],[2],[3],[4]]=>0
[[1,2,3,4,5]]=>5
[[1,3,4,5],[2]]=>0
[[1,2,4,5],[3]]=>1
[[1,2,3,5],[4]]=>2
[[1,2,3,4],[5]]=>3
[[1,3,5],[2,4]]=>0
[[1,2,5],[3,4]]=>1
[[1,3,4],[2,5]]=>0
[[1,2,4],[3,5]]=>1
[[1,2,3],[4,5]]=>2
[[1,4,5],[2],[3]]=>1
[[1,3,5],[2],[4]]=>0
[[1,2,5],[3],[4]]=>2
[[1,3,4],[2],[5]]=>0
[[1,2,4],[3],[5]]=>1
[[1,2,3],[4],[5]]=>3
[[1,4],[2,5],[3]]=>1
[[1,3],[2,5],[4]]=>0
[[1,2],[3,5],[4]]=>2
[[1,3],[2,4],[5]]=>0
[[1,2],[3,4],[5]]=>1
[[1,5],[2],[3],[4]]=>0
[[1,4],[2],[3],[5]]=>1
[[1,3],[2],[4],[5]]=>0
[[1,2],[3],[4],[5]]=>1
[[1],[2],[3],[4],[5]]=>1
[[1,2,3,4,5,6]]=>6
[[1,3,4,5,6],[2]]=>0
[[1,2,4,5,6],[3]]=>1
[[1,2,3,5,6],[4]]=>2
[[1,2,3,4,6],[5]]=>3
[[1,2,3,4,5],[6]]=>4
[[1,3,5,6],[2,4]]=>0
[[1,2,5,6],[3,4]]=>1
[[1,3,4,6],[2,5]]=>0
[[1,2,4,6],[3,5]]=>1
[[1,2,3,6],[4,5]]=>2
[[1,3,4,5],[2,6]]=>0
[[1,2,4,5],[3,6]]=>1
[[1,2,3,5],[4,6]]=>2
[[1,2,3,4],[5,6]]=>3
[[1,4,5,6],[2],[3]]=>1
[[1,3,5,6],[2],[4]]=>0
[[1,2,5,6],[3],[4]]=>2
[[1,3,4,6],[2],[5]]=>0
[[1,2,4,6],[3],[5]]=>1
[[1,2,3,6],[4],[5]]=>3
[[1,3,4,5],[2],[6]]=>0
[[1,2,4,5],[3],[6]]=>1
[[1,2,3,5],[4],[6]]=>2
[[1,2,3,4],[5],[6]]=>4
[[1,3,5],[2,4,6]]=>0
[[1,2,5],[3,4,6]]=>1
[[1,3,4],[2,5,6]]=>0
[[1,2,4],[3,5,6]]=>1
[[1,2,3],[4,5,6]]=>2
[[1,4,6],[2,5],[3]]=>1
[[1,3,6],[2,5],[4]]=>0
[[1,2,6],[3,5],[4]]=>2
[[1,3,6],[2,4],[5]]=>0
[[1,2,6],[3,4],[5]]=>1
[[1,4,5],[2,6],[3]]=>1
[[1,3,5],[2,6],[4]]=>0
[[1,2,5],[3,6],[4]]=>2
[[1,3,4],[2,6],[5]]=>0
[[1,2,4],[3,6],[5]]=>1
[[1,2,3],[4,6],[5]]=>3
[[1,3,5],[2,4],[6]]=>0
[[1,2,5],[3,4],[6]]=>1
[[1,3,4],[2,5],[6]]=>0
[[1,2,4],[3,5],[6]]=>1
[[1,2,3],[4,5],[6]]=>2
[[1,5,6],[2],[3],[4]]=>0
[[1,4,6],[2],[3],[5]]=>1
[[1,3,6],[2],[4],[5]]=>0
[[1,2,6],[3],[4],[5]]=>1
[[1,4,5],[2],[3],[6]]=>1
[[1,3,5],[2],[4],[6]]=>0
[[1,2,5],[3],[4],[6]]=>2
[[1,3,4],[2],[5],[6]]=>0
[[1,2,4],[3],[5],[6]]=>1
[[1,2,3],[4],[5],[6]]=>2
[[1,4],[2,5],[3,6]]=>1
[[1,3],[2,5],[4,6]]=>0
[[1,2],[3,5],[4,6]]=>2
[[1,3],[2,4],[5,6]]=>0
[[1,2],[3,4],[5,6]]=>1
[[1,5],[2,6],[3],[4]]=>0
[[1,4],[2,6],[3],[5]]=>1
[[1,3],[2,6],[4],[5]]=>0
[[1,2],[3,6],[4],[5]]=>1
[[1,4],[2,5],[3],[6]]=>1
[[1,3],[2,5],[4],[6]]=>0
[[1,2],[3,5],[4],[6]]=>2
[[1,3],[2,4],[5],[6]]=>0
[[1,2],[3,4],[5],[6]]=>1
[[1,6],[2],[3],[4],[5]]=>1
[[1,5],[2],[3],[4],[6]]=>0
[[1,4],[2],[3],[5],[6]]=>1
[[1,3],[2],[4],[5],[6]]=>0
[[1,2],[3],[4],[5],[6]]=>2
[[1],[2],[3],[4],[5],[6]]=>0
[[1,2,3,4,5,6,7]]=>7
[[1,3,4,5,6,7],[2]]=>0
[[1,2,4,5,6,7],[3]]=>1
[[1,2,3,5,6,7],[4]]=>2
[[1,2,3,4,6,7],[5]]=>3
[[1,2,3,4,5,7],[6]]=>4
[[1,2,3,4,5,6],[7]]=>5
[[1,3,5,6,7],[2,4]]=>0
[[1,2,5,6,7],[3,4]]=>1
[[1,3,4,6,7],[2,5]]=>0
[[1,2,4,6,7],[3,5]]=>1
[[1,2,3,6,7],[4,5]]=>2
[[1,3,4,5,7],[2,6]]=>0
[[1,2,4,5,7],[3,6]]=>1
[[1,2,3,5,7],[4,6]]=>2
[[1,2,3,4,7],[5,6]]=>3
[[1,3,4,5,6],[2,7]]=>0
[[1,2,4,5,6],[3,7]]=>1
[[1,2,3,5,6],[4,7]]=>2
[[1,2,3,4,6],[5,7]]=>3
[[1,2,3,4,5],[6,7]]=>4
[[1,4,5,6,7],[2],[3]]=>1
[[1,3,5,6,7],[2],[4]]=>0
[[1,2,5,6,7],[3],[4]]=>2
[[1,3,4,6,7],[2],[5]]=>0
[[1,2,4,6,7],[3],[5]]=>1
[[1,2,3,6,7],[4],[5]]=>3
[[1,3,4,5,7],[2],[6]]=>0
[[1,2,4,5,7],[3],[6]]=>1
[[1,2,3,5,7],[4],[6]]=>2
[[1,2,3,4,7],[5],[6]]=>4
[[1,3,4,5,6],[2],[7]]=>0
[[1,2,4,5,6],[3],[7]]=>1
[[1,2,3,5,6],[4],[7]]=>2
[[1,2,3,4,6],[5],[7]]=>3
[[1,2,3,4,5],[6],[7]]=>5
[[1,3,5,7],[2,4,6]]=>0
[[1,2,5,7],[3,4,6]]=>1
[[1,3,4,7],[2,5,6]]=>0
[[1,2,4,7],[3,5,6]]=>1
[[1,2,3,7],[4,5,6]]=>2
[[1,3,5,6],[2,4,7]]=>0
[[1,2,5,6],[3,4,7]]=>1
[[1,3,4,6],[2,5,7]]=>0
[[1,2,4,6],[3,5,7]]=>1
[[1,2,3,6],[4,5,7]]=>2
[[1,3,4,5],[2,6,7]]=>0
[[1,2,4,5],[3,6,7]]=>1
[[1,2,3,5],[4,6,7]]=>2
[[1,2,3,4],[5,6,7]]=>3
[[1,4,6,7],[2,5],[3]]=>1
[[1,3,6,7],[2,5],[4]]=>0
[[1,2,6,7],[3,5],[4]]=>2
[[1,3,6,7],[2,4],[5]]=>0
[[1,2,6,7],[3,4],[5]]=>1
[[1,4,5,7],[2,6],[3]]=>1
[[1,3,5,7],[2,6],[4]]=>0
[[1,2,5,7],[3,6],[4]]=>2
[[1,3,4,7],[2,6],[5]]=>0
[[1,2,4,7],[3,6],[5]]=>1
[[1,2,3,7],[4,6],[5]]=>3
[[1,3,5,7],[2,4],[6]]=>0
[[1,2,5,7],[3,4],[6]]=>1
[[1,3,4,7],[2,5],[6]]=>0
[[1,2,4,7],[3,5],[6]]=>1
[[1,2,3,7],[4,5],[6]]=>2
[[1,4,5,6],[2,7],[3]]=>1
[[1,3,5,6],[2,7],[4]]=>0
[[1,2,5,6],[3,7],[4]]=>2
[[1,3,4,6],[2,7],[5]]=>0
[[1,2,4,6],[3,7],[5]]=>1
[[1,2,3,6],[4,7],[5]]=>3
[[1,3,4,5],[2,7],[6]]=>0
[[1,2,4,5],[3,7],[6]]=>1
[[1,2,3,5],[4,7],[6]]=>2
[[1,2,3,4],[5,7],[6]]=>4
[[1,3,5,6],[2,4],[7]]=>0
[[1,2,5,6],[3,4],[7]]=>1
[[1,3,4,6],[2,5],[7]]=>0
[[1,2,4,6],[3,5],[7]]=>1
[[1,2,3,6],[4,5],[7]]=>2
[[1,3,4,5],[2,6],[7]]=>0
[[1,2,4,5],[3,6],[7]]=>1
[[1,2,3,5],[4,6],[7]]=>2
[[1,2,3,4],[5,6],[7]]=>3
[[1,5,6,7],[2],[3],[4]]=>0
[[1,4,6,7],[2],[3],[5]]=>1
[[1,3,6,7],[2],[4],[5]]=>0
[[1,2,6,7],[3],[4],[5]]=>1
[[1,4,5,7],[2],[3],[6]]=>1
[[1,3,5,7],[2],[4],[6]]=>0
[[1,2,5,7],[3],[4],[6]]=>2
[[1,3,4,7],[2],[5],[6]]=>0
[[1,2,4,7],[3],[5],[6]]=>1
[[1,2,3,7],[4],[5],[6]]=>2
[[1,4,5,6],[2],[3],[7]]=>1
[[1,3,5,6],[2],[4],[7]]=>0
[[1,2,5,6],[3],[4],[7]]=>2
[[1,3,4,6],[2],[5],[7]]=>0
[[1,2,4,6],[3],[5],[7]]=>1
[[1,2,3,6],[4],[5],[7]]=>3
[[1,3,4,5],[2],[6],[7]]=>0
[[1,2,4,5],[3],[6],[7]]=>1
[[1,2,3,5],[4],[6],[7]]=>2
[[1,2,3,4],[5],[6],[7]]=>3
[[1,4,6],[2,5,7],[3]]=>1
[[1,3,6],[2,5,7],[4]]=>0
[[1,2,6],[3,5,7],[4]]=>2
[[1,3,6],[2,4,7],[5]]=>0
[[1,2,6],[3,4,7],[5]]=>1
[[1,4,5],[2,6,7],[3]]=>1
[[1,3,5],[2,6,7],[4]]=>0
[[1,2,5],[3,6,7],[4]]=>2
[[1,3,4],[2,6,7],[5]]=>0
[[1,2,4],[3,6,7],[5]]=>1
[[1,2,3],[4,6,7],[5]]=>3
[[1,3,5],[2,4,7],[6]]=>0
[[1,2,5],[3,4,7],[6]]=>1
[[1,3,4],[2,5,7],[6]]=>0
[[1,2,4],[3,5,7],[6]]=>1
[[1,2,3],[4,5,7],[6]]=>2
[[1,3,5],[2,4,6],[7]]=>0
[[1,2,5],[3,4,6],[7]]=>1
[[1,3,4],[2,5,6],[7]]=>0
[[1,2,4],[3,5,6],[7]]=>1
[[1,2,3],[4,5,6],[7]]=>2
[[1,4,7],[2,5],[3,6]]=>1
[[1,3,7],[2,5],[4,6]]=>0
[[1,2,7],[3,5],[4,6]]=>2
[[1,3,7],[2,4],[5,6]]=>0
[[1,2,7],[3,4],[5,6]]=>1
[[1,4,6],[2,5],[3,7]]=>1
[[1,3,6],[2,5],[4,7]]=>0
[[1,2,6],[3,5],[4,7]]=>2
[[1,3,6],[2,4],[5,7]]=>0
[[1,2,6],[3,4],[5,7]]=>1
[[1,4,5],[2,6],[3,7]]=>1
[[1,3,5],[2,6],[4,7]]=>0
[[1,2,5],[3,6],[4,7]]=>2
[[1,3,4],[2,6],[5,7]]=>0
[[1,2,4],[3,6],[5,7]]=>1
[[1,2,3],[4,6],[5,7]]=>3
[[1,3,5],[2,4],[6,7]]=>0
[[1,2,5],[3,4],[6,7]]=>1
[[1,3,4],[2,5],[6,7]]=>0
[[1,2,4],[3,5],[6,7]]=>1
[[1,2,3],[4,5],[6,7]]=>2
[[1,5,7],[2,6],[3],[4]]=>0
[[1,4,7],[2,6],[3],[5]]=>1
[[1,3,7],[2,6],[4],[5]]=>0
[[1,2,7],[3,6],[4],[5]]=>1
[[1,4,7],[2,5],[3],[6]]=>1
[[1,3,7],[2,5],[4],[6]]=>0
[[1,2,7],[3,5],[4],[6]]=>2
[[1,3,7],[2,4],[5],[6]]=>0
[[1,2,7],[3,4],[5],[6]]=>1
[[1,5,6],[2,7],[3],[4]]=>0
[[1,4,6],[2,7],[3],[5]]=>1
[[1,3,6],[2,7],[4],[5]]=>0
[[1,2,6],[3,7],[4],[5]]=>1
[[1,4,5],[2,7],[3],[6]]=>1
[[1,3,5],[2,7],[4],[6]]=>0
[[1,2,5],[3,7],[4],[6]]=>2
[[1,3,4],[2,7],[5],[6]]=>0
[[1,2,4],[3,7],[5],[6]]=>1
[[1,2,3],[4,7],[5],[6]]=>2
[[1,4,6],[2,5],[3],[7]]=>1
[[1,3,6],[2,5],[4],[7]]=>0
[[1,2,6],[3,5],[4],[7]]=>2
[[1,3,6],[2,4],[5],[7]]=>0
[[1,2,6],[3,4],[5],[7]]=>1
[[1,4,5],[2,6],[3],[7]]=>1
[[1,3,5],[2,6],[4],[7]]=>0
[[1,2,5],[3,6],[4],[7]]=>2
[[1,3,4],[2,6],[5],[7]]=>0
[[1,2,4],[3,6],[5],[7]]=>1
[[1,2,3],[4,6],[5],[7]]=>3
[[1,3,5],[2,4],[6],[7]]=>0
[[1,2,5],[3,4],[6],[7]]=>1
[[1,3,4],[2,5],[6],[7]]=>0
[[1,2,4],[3,5],[6],[7]]=>1
[[1,2,3],[4,5],[6],[7]]=>2
[[1,6,7],[2],[3],[4],[5]]=>1
[[1,5,7],[2],[3],[4],[6]]=>0
[[1,4,7],[2],[3],[5],[6]]=>1
[[1,3,7],[2],[4],[5],[6]]=>0
[[1,2,7],[3],[4],[5],[6]]=>2
[[1,5,6],[2],[3],[4],[7]]=>0
[[1,4,6],[2],[3],[5],[7]]=>1
[[1,3,6],[2],[4],[5],[7]]=>0
[[1,2,6],[3],[4],[5],[7]]=>1
[[1,4,5],[2],[3],[6],[7]]=>1
[[1,3,5],[2],[4],[6],[7]]=>0
[[1,2,5],[3],[4],[6],[7]]=>2
[[1,3,4],[2],[5],[6],[7]]=>0
[[1,2,4],[3],[5],[6],[7]]=>1
[[1,2,3],[4],[5],[6],[7]]=>3
[[1,5],[2,6],[3,7],[4]]=>0
[[1,4],[2,6],[3,7],[5]]=>1
[[1,3],[2,6],[4,7],[5]]=>0
[[1,2],[3,6],[4,7],[5]]=>1
[[1,4],[2,5],[3,7],[6]]=>1
[[1,3],[2,5],[4,7],[6]]=>0
[[1,2],[3,5],[4,7],[6]]=>2
[[1,3],[2,4],[5,7],[6]]=>0
[[1,2],[3,4],[5,7],[6]]=>1
[[1,4],[2,5],[3,6],[7]]=>1
[[1,3],[2,5],[4,6],[7]]=>0
[[1,2],[3,5],[4,6],[7]]=>2
[[1,3],[2,4],[5,6],[7]]=>0
[[1,2],[3,4],[5,6],[7]]=>1
[[1,6],[2,7],[3],[4],[5]]=>1
[[1,5],[2,7],[3],[4],[6]]=>0
[[1,4],[2,7],[3],[5],[6]]=>1
[[1,3],[2,7],[4],[5],[6]]=>0
[[1,2],[3,7],[4],[5],[6]]=>2
[[1,5],[2,6],[3],[4],[7]]=>0
[[1,4],[2,6],[3],[5],[7]]=>1
[[1,3],[2,6],[4],[5],[7]]=>0
[[1,2],[3,6],[4],[5],[7]]=>1
[[1,4],[2,5],[3],[6],[7]]=>1
[[1,3],[2,5],[4],[6],[7]]=>0
[[1,2],[3,5],[4],[6],[7]]=>2
[[1,3],[2,4],[5],[6],[7]]=>0
[[1,2],[3,4],[5],[6],[7]]=>1
[[1,7],[2],[3],[4],[5],[6]]=>0
[[1,6],[2],[3],[4],[5],[7]]=>1
[[1,5],[2],[3],[4],[6],[7]]=>0
[[1,4],[2],[3],[5],[6],[7]]=>1
[[1,3],[2],[4],[5],[6],[7]]=>0
[[1,2],[3],[4],[5],[6],[7]]=>1
[[1],[2],[3],[4],[5],[6],[7]]=>1
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Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is St000508Eigenvalues of the random-to-random operator acting on a simple module..
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is St000508Eigenvalues of the random-to-random operator acting on a simple module..
References
[1] Garsia, A. M., Wallach, N. $r$-Qsym is free over Sym MathSciNet:2319171
[2] Lafrenière, N. Eigenvalues of symmetrized shuffling operators MathSciNet:4098299
[3] Reiner, V., Saliola, F., Welker, V. Spectra of symmetrized shuffling operators MathSciNet:3184410
[2] Lafrenière, N. Eigenvalues of symmetrized shuffling operators MathSciNet:4098299
[3] Reiner, V., Saliola, F., Welker, V. Spectra of symmetrized shuffling operators MathSciNet:3184410
Code
def statistic(T):
if T.height() <= 1:
return T.size()
r = 0
for i in range(1, T.height()-1):
if T[i+1][0] != T[i][0]+1:
break
r += 1
j = T[1][0]-2 if r % 2 == 0 else T[1][0]-1
return j
Created
Jul 13, 2022 at 17:16 by Nadia Lafreniere
Updated
Jul 13, 2022 at 17:16 by Nadia Lafreniere
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