Identifier
-
Mp00084:
Standard tableaux
—conjugate⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000337: Permutations ⟶ ℤ
Values
[[1]] => [[1]] => [1] => [1] => 0
[[1,2]] => [[1],[2]] => [2,1] => [2,1] => 1
[[1],[2]] => [[1,2]] => [1,2] => [1,2] => 0
[[1,2,3]] => [[1],[2],[3]] => [3,2,1] => [2,3,1] => 1
[[1,3],[2]] => [[1,2],[3]] => [3,1,2] => [3,1,2] => 2
[[1,2],[3]] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]] => [[1],[2],[3],[4]] => [4,3,2,1] => [3,2,4,1] => 2
[[1,3,4],[2]] => [[1,2],[3],[4]] => [4,3,1,2] => [3,1,4,2] => 2
[[1,2,4],[3]] => [[1,3],[2],[4]] => [4,2,1,3] => [2,4,1,3] => 2
[[1,2,3],[4]] => [[1,4],[2],[3]] => [3,2,1,4] => [2,3,1,4] => 1
[[1,3],[2,4]] => [[1,2],[3,4]] => [3,4,1,2] => [4,1,3,2] => 2
[[1,2],[3,4]] => [[1,3],[2,4]] => [2,4,1,3] => [4,2,1,3] => 1
[[1,4],[2],[3]] => [[1,2,3],[4]] => [4,1,2,3] => [4,1,2,3] => 3
[[1,3],[2],[4]] => [[1,2,4],[3]] => [3,1,2,4] => [3,1,2,4] => 2
[[1,2],[3],[4]] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 1
[[1],[2],[3],[4]] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[[1,3,4,5],[2]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [3,4,1,5,2] => 2
[[1,2,4,5],[3]] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [4,2,5,1,3] => 3
[[1,2,3,5],[4]] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [3,2,5,1,4] => 3
[[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [3,2,4,1,5] => 2
[[1,3,5],[2,4]] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,3,1,5,2] => 2
[[1,2,5],[3,4]] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [2,4,1,5,3] => 2
[[1,3,4],[2,5]] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [5,3,1,4,2] => 2
[[1,2,4],[3,5]] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,5,1,4,3] => 2
[[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [2,5,3,1,4] => 1
[[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [4,1,5,2,3] => 3
[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,1,5,2,4] => 3
[[1,2,5],[3],[4]] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,5,1,3,4] => 3
[[1,3,4],[2],[5]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,1,4,2,5] => 2
[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,1,3,5] => 2
[[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[[1,4],[2,5],[3]] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [5,1,4,2,3] => 3
[[1,3],[2,5],[4]] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [5,1,3,2,4] => 2
[[1,2],[3,5],[4]] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [5,2,1,3,4] => 1
[[1,3],[2,4],[5]] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [4,1,3,2,5] => 2
[[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[[1,5],[2],[3],[4]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[[1,4],[2],[3],[5]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [4,1,2,3,5] => 3
[[1,3],[2],[4],[5]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [3,1,2,4,5] => 2
[[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 3
[[1,3,4,5,6],[2]] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [4,3,5,1,6,2] => 3
[[1,2,4,5,6],[3]] => [[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [4,2,5,1,6,3] => 3
[[1,2,3,5,6],[4]] => [[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [3,5,2,6,1,4] => 3
[[1,2,3,4,6],[5]] => [[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [3,4,2,6,1,5] => 3
[[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [3,4,2,5,1,6] => 2
[[1,3,5,6],[2,4]] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [3,4,5,1,6,2] => 2
[[1,2,5,6],[3,4]] => [[1,3],[2,4],[5],[6]] => [6,5,2,4,1,3] => [4,5,2,6,1,3] => 3
[[1,3,4,6],[2,5]] => [[1,2],[3,5],[4],[6]] => [6,4,3,5,1,2] => [3,5,4,1,6,2] => 2
[[1,2,4,6],[3,5]] => [[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [5,4,2,6,1,3] => 3
[[1,2,3,6],[4,5]] => [[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [3,2,5,1,6,4] => 3
[[1,3,4,5],[2,6]] => [[1,2],[3,6],[4],[5]] => [5,4,3,6,1,2] => [3,6,4,1,5,2] => 2
[[1,2,4,5],[3,6]] => [[1,3],[2,6],[4],[5]] => [5,4,2,6,1,3] => [6,4,2,5,1,3] => 3
[[1,2,3,5],[4,6]] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [3,2,6,1,5,4] => 3
[[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [3,2,6,4,1,5] => 2
[[1,4,5,6],[2],[3]] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [4,1,5,2,6,3] => 3
[[1,3,5,6],[2],[4]] => [[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [3,5,1,6,2,4] => 3
[[1,2,5,6],[3],[4]] => [[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [5,2,6,1,3,4] => 4
[[1,3,4,6],[2],[5]] => [[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [3,4,1,6,2,5] => 3
[[1,2,4,6],[3],[5]] => [[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [4,2,6,1,3,5] => 4
[[1,2,3,6],[4],[5]] => [[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => 4
[[1,3,4,5],[2],[6]] => [[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [3,4,1,5,2,6] => 2
[[1,2,4,5],[3],[6]] => [[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [4,2,5,1,3,6] => 3
[[1,2,3,5],[4],[6]] => [[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [3,2,5,1,4,6] => 3
[[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [3,2,4,1,5,6] => 2
[[1,3,5],[2,4,6]] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [3,4,6,1,5,2] => 2
[[1,2,5],[3,4,6]] => [[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [4,6,2,5,1,3] => 3
[[1,3,4],[2,5,6]] => [[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [3,5,1,6,4,2] => 2
[[1,2,4],[3,5,6]] => [[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [5,2,6,4,1,3] => 3
[[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [5,2,3,1,6,4] => 3
[[1,4,6],[2,5],[3]] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [5,1,4,2,6,3] => 3
[[1,3,6],[2,5],[4]] => [[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [5,3,1,6,2,4] => 3
[[1,2,6],[3,5],[4]] => [[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,5,1,6,3,4] => 3
[[1,3,6],[2,4],[5]] => [[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [4,3,1,6,2,5] => 3
[[1,2,6],[3,4],[5]] => [[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,4,1,6,3,5] => 3
[[1,4,5],[2,6],[3]] => [[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [6,1,4,2,5,3] => 3
[[1,3,5],[2,6],[4]] => [[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [6,3,1,5,2,4] => 3
[[1,2,5],[3,6],[4]] => [[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,6,1,5,3,4] => 3
[[1,3,4],[2,6],[5]] => [[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [6,3,1,4,2,5] => 2
[[1,2,4],[3,6],[5]] => [[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [2,6,1,4,3,5] => 2
[[1,2,3],[4,6],[5]] => [[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [2,6,3,1,4,5] => 1
[[1,3,5],[2,4],[6]] => [[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [4,3,1,5,2,6] => 2
[[1,2,5],[3,4],[6]] => [[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,4,1,5,3,6] => 2
[[1,3,4],[2,5],[6]] => [[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [5,3,1,4,2,6] => 2
[[1,2,4],[3,5],[6]] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,5,1,4,3,6] => 2
[[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [2,5,3,1,4,6] => 1
[[1,5,6],[2],[3],[4]] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [5,1,6,2,3,4] => 4
[[1,4,6],[2],[3],[5]] => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [4,1,6,2,3,5] => 4
[[1,3,6],[2],[4],[5]] => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 4
[[1,2,6],[3],[4],[5]] => [[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [2,6,1,3,4,5] => 4
[[1,4,5],[2],[3],[6]] => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [4,1,5,2,3,6] => 3
[[1,3,5],[2],[4],[6]] => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [3,1,5,2,4,6] => 3
[[1,2,5],[3],[4],[6]] => [[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,5,1,3,4,6] => 3
[[1,3,4],[2],[5],[6]] => [[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [3,1,4,2,5,6] => 2
[[1,2,4],[3],[5],[6]] => [[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => 2
[[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [2,3,1,4,5,6] => 1
[[1,4],[2,5],[3,6]] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [6,1,5,2,4,3] => 3
[[1,3],[2,5],[4,6]] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [6,1,5,3,2,4] => 3
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Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.
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