Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001816
(load all 3 compositions to match this statistic)
Mapping
St001816: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[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
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 [[St000508]].
Matching statistic: St000338
(load all 6 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000338: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 88%
Values
[[1]]
 => [1] => [1] => 1
[[1,2]]
 => [1,2] => [1,2] => 2
[[1],[2]]
 => [2,1] => [2,1] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => 3
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => 0
[[1,2],[3]]
 => [3,1,2] => [2,3,1] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 4
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 0
[[1,2,4],[3]]
 => [3,1,2,4] => [2,3,1,4] => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,3,4,1] => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => 0
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,2,4,1] => 0
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,3,1,4,5] => 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,3,4,1,5] => 2
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [2,3,4,5,1] => 3
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [3,1,4,2,5] => 0
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2,5] => 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [3,1,4,5,2] => 0
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,4,1,5,2] => 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,4,5,1,2] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [3,2,4,1,5] => 0
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [3,4,2,1,5] => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [3,2,4,5,1] => 0
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [3,4,2,5,1] => 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,4,5,2,1] => 3
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [4,2,1,5,3] => 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => 0
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,2,1,3] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [4,2,5,3,1] => 0
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [4,5,2,3,1] => 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => 0
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,2,5,1] => 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,3,5,2,1] => 0
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [4,5,3,2,1] => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [2,3,1,4,5,6] => 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [2,3,4,1,5,6] => 2
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 3
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 4
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [3,1,4,2,5,6] => 0
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 0
[[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 1
[[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 2
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 3
[[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 4
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 5
[[1,3,5,6,7],[2,4]]
 => [2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => ? = 0
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,4,1,2,5,6,7] => ? = 1
[[1,3,4,6,7],[2,5]]
 => [2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => ? = 0
[[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => [3,4,1,5,2,6,7] => ? = 1
[[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [3,4,5,1,2,6,7] => ? = 2
[[1,3,4,5,7],[2,6]]
 => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 0
[[1,2,4,5,7],[3,6]]
 => [3,6,1,2,4,5,7] => [3,4,1,5,6,2,7] => ? = 1
[[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 2
[[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 3
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 0
[[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => [3,4,1,5,6,7,2] => ? = 1
[[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => ? = 2
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 3
[[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => ? = 4
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 1
[[1,3,5,6,7],[2],[4]]
 => [4,2,1,3,5,6,7] => [3,2,4,1,5,6,7] => ? = 0
[[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [3,4,2,1,5,6,7] => ? = 2
[[1,3,4,6,7],[2],[5]]
 => [5,2,1,3,4,6,7] => [3,2,4,5,1,6,7] => ? = 0
[[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => ? = 1
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [3,4,5,2,1,6,7] => ? = 3
[[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => [3,2,4,5,6,1,7] => ? = 0
[[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => [3,4,2,5,6,1,7] => ? = 1
[[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => [3,4,5,2,6,1,7] => ? = 2
[[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => ? = 4
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => ? = 0
[[1,2,4,5,6],[3],[7]]
 => [7,3,1,2,4,5,6] => [3,4,2,5,6,7,1] => ? = 1
[[1,2,3,5,6],[4],[7]]
 => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => ? = 2
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 3
[[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 5
[[1,3,5,7],[2,4,6]]
 => [2,4,6,1,3,5,7] => [4,1,5,2,6,3,7] => ? = 0
[[1,2,5,7],[3,4,6]]
 => [3,4,6,1,2,5,7] => [4,5,1,2,6,3,7] => ? = 1
[[1,3,4,7],[2,5,6]]
 => [2,5,6,1,3,4,7] => [4,1,5,6,2,3,7] => ? = 0
[[1,2,4,7],[3,5,6]]
 => [3,5,6,1,2,4,7] => [4,5,1,6,2,3,7] => ? = 1
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 2
[[1,3,5,6],[2,4,7]]
 => [2,4,7,1,3,5,6] => [4,1,5,2,6,7,3] => ? = 0
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [4,5,1,2,6,7,3] => ? = 1
[[1,3,4,6],[2,5,7]]
 => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => ? = 0
[[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => [4,5,1,6,2,7,3] => ? = 1
[[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 2
[[1,3,4,5],[2,6,7]]
 => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 0
[[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => [4,5,1,6,7,2,3] => ? = 1
[[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => ? = 2
[[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 3
Description
The number of pixed points of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$\textrm{pix} \, \sigma = \textrm{length} (p)$$.