Identifier
Values
[1] => [[1]] => 1
[2] => [[1,2]] => 2
[1,1] => [[1],[2]] => 0
[3] => [[1,2,3]] => 3
[2,1] => [[1,2],[3]] => 1
[1,1,1] => [[1],[2],[3]] => 1
[4] => [[1,2,3,4]] => 4
[3,1] => [[1,2,3],[4]] => 2
[2,2] => [[1,2],[3,4]] => 1
[2,1,1] => [[1,2],[3],[4]] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => 0
[5] => [[1,2,3,4,5]] => 5
[4,1] => [[1,2,3,4],[5]] => 3
[3,2] => [[1,2,3],[4,5]] => 2
[3,1,1] => [[1,2,3],[4],[5]] => 3
[2,2,1] => [[1,2],[3,4],[5]] => 1
[2,1,1,1] => [[1,2],[3],[4],[5]] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 1
[6] => [[1,2,3,4,5,6]] => 6
[5,1] => [[1,2,3,4,5],[6]] => 4
[4,2] => [[1,2,3,4],[5,6]] => 3
[4,1,1] => [[1,2,3,4],[5],[6]] => 4
[3,3] => [[1,2,3],[4,5,6]] => 2
[3,2,1] => [[1,2,3],[4,5],[6]] => 2
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => 2
[2,2,2] => [[1,2],[3,4],[5,6]] => 1
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => 1
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 0
[7] => [[1,2,3,4,5,6,7]] => 7
[6,1] => [[1,2,3,4,5,6],[7]] => 5
[5,2] => [[1,2,3,4,5],[6,7]] => 4
[5,1,1] => [[1,2,3,4,5],[6],[7]] => 5
[4,3] => [[1,2,3,4],[5,6,7]] => 3
[4,2,1] => [[1,2,3,4],[5,6],[7]] => 3
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => 3
[3,3,1] => [[1,2,3],[4,5,6],[7]] => 2
[3,2,2] => [[1,2,3],[4,5],[6,7]] => 2
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => 2
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => 3
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => 1
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => 1
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => 1
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => 1
[] => [] => 0
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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..
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.