Identifier
-
Mp00084:
Standard tableaux
—conjugate⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000834: Permutations ⟶ ℤ
Values
[[1]] => [[1]] => [1] => [1] => 0
[[1,2]] => [[1],[2]] => [2,1] => [2,1] => 0
[[1],[2]] => [[1,2]] => [1,2] => [1,2] => 1
[[1,2,3]] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 0
[[1,3],[2]] => [[1,2],[3]] => [3,1,2] => [2,3,1] => 1
[[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] => 1
[[1,2,3,4]] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 0
[[1,3,4],[2]] => [[1,2],[3],[4]] => [4,3,1,2] => [2,4,3,1] => 1
[[1,2,4],[3]] => [[1,3],[2],[4]] => [4,2,1,3] => [3,4,2,1] => 1
[[1,2,3],[4]] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 1
[[1,3],[2,4]] => [[1,2],[3,4]] => [3,4,1,2] => [2,4,1,3] => 2
[[1,2],[3,4]] => [[1,3],[2,4]] => [2,4,1,3] => [3,4,1,2] => 2
[[1,4],[2],[3]] => [[1,2,3],[4]] => [4,1,2,3] => [2,3,4,1] => 1
[[1,3],[2],[4]] => [[1,2,4],[3]] => [3,1,2,4] => [2,3,1,4] => 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] => 1
[[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,5,4,3,1] => 1
[[1,2,4,5],[3]] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [3,5,4,2,1] => 1
[[1,2,3,5],[4]] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,5,3,2,1] => 1
[[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[[1,3,5],[2,4]] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,4,1,5,3] => 2
[[1,2,5],[3,4]] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [3,4,1,5,2] => 2
[[1,3,4],[2,5]] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,5,1,4,3] => 2
[[1,2,4],[3,5]] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,5,1,4,2] => 2
[[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,5,1,3,2] => 2
[[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [2,3,5,4,1] => 1
[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [2,4,5,3,1] => 1
[[1,2,5],[3],[4]] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,4,5,2,1] => 1
[[1,3,4],[2],[5]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [2,4,3,1,5] => 2
[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,4,2,1,5] => 2
[[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[[1,4],[2,5],[3]] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [2,3,5,1,4] => 2
[[1,3],[2,5],[4]] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [2,4,5,1,3] => 2
[[1,2],[3,5],[4]] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [3,4,5,1,2] => 2
[[1,3],[2,4],[5]] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [2,4,1,3,5] => 2
[[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [3,4,1,2,5] => 2
[[1,5],[2],[3],[4]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[[1,4],[2],[3],[5]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[[1,3],[2],[4],[5]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [2,3,1,4,5] => 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] => 1
[[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [2,6,5,4,3,1] => 1
[[1,2,4,5,6],[3]] => [[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [3,6,5,4,2,1] => 1
[[1,2,3,5,6],[4]] => [[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [4,6,5,3,2,1] => 1
[[1,2,3,4,6],[5]] => [[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [5,6,4,3,2,1] => 1
[[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[[1,3,5,6],[2,4]] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [2,4,1,6,5,3] => 2
[[1,2,5,6],[3,4]] => [[1,3],[2,4],[5],[6]] => [6,5,2,4,1,3] => [3,4,1,6,5,2] => 2
[[1,3,4,6],[2,5]] => [[1,2],[3,5],[4],[6]] => [6,4,3,5,1,2] => [2,5,1,6,4,3] => 2
[[1,2,4,6],[3,5]] => [[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [3,5,1,6,4,2] => 2
[[1,2,3,6],[4,5]] => [[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [4,5,1,6,3,2] => 2
[[1,3,4,5],[2,6]] => [[1,2],[3,6],[4],[5]] => [5,4,3,6,1,2] => [2,6,1,5,4,3] => 2
[[1,2,4,5],[3,6]] => [[1,3],[2,6],[4],[5]] => [5,4,2,6,1,3] => [3,6,1,5,4,2] => 2
[[1,2,3,5],[4,6]] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [4,6,1,5,3,2] => 2
[[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [5,6,1,4,3,2] => 2
[[1,4,5,6],[2],[3]] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [2,3,6,5,4,1] => 1
[[1,3,5,6],[2],[4]] => [[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [2,4,6,5,3,1] => 1
[[1,2,5,6],[3],[4]] => [[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [3,4,6,5,2,1] => 1
[[1,3,4,6],[2],[5]] => [[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [2,5,6,4,3,1] => 1
[[1,2,4,6],[3],[5]] => [[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [3,5,6,4,2,1] => 1
[[1,2,3,6],[4],[5]] => [[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [4,5,6,3,2,1] => 1
[[1,3,4,5],[2],[6]] => [[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [2,5,4,3,1,6] => 2
[[1,2,4,5],[3],[6]] => [[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [3,5,4,2,1,6] => 2
[[1,2,3,5],[4],[6]] => [[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [4,5,3,2,1,6] => 2
[[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 1
[[1,3,5],[2,4,6]] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,4,1,6,3,5] => 3
[[1,2,5],[3,4,6]] => [[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [3,4,1,6,2,5] => 3
[[1,3,4],[2,5,6]] => [[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [2,5,1,6,3,4] => 3
[[1,2,4],[3,5,6]] => [[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,5,1,6,2,4] => 3
[[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,5,1,6,2,3] => 3
[[1,4,6],[2,5],[3]] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [2,3,5,1,6,4] => 2
[[1,3,6],[2,5],[4]] => [[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [2,4,5,1,6,3] => 2
[[1,2,6],[3,5],[4]] => [[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [3,4,5,1,6,2] => 2
[[1,3,6],[2,4],[5]] => [[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [2,4,1,5,6,3] => 2
[[1,2,6],[3,4],[5]] => [[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [3,4,1,5,6,2] => 2
[[1,4,5],[2,6],[3]] => [[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [2,3,6,1,5,4] => 2
[[1,3,5],[2,6],[4]] => [[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [2,4,6,1,5,3] => 2
[[1,2,5],[3,6],[4]] => [[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [3,4,6,1,5,2] => 2
[[1,3,4],[2,6],[5]] => [[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [2,5,6,1,4,3] => 2
[[1,2,4],[3,6],[5]] => [[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [3,5,6,1,4,2] => 2
[[1,2,3],[4,6],[5]] => [[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [4,5,6,1,3,2] => 2
[[1,3,5],[2,4],[6]] => [[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [2,4,1,5,3,6] => 3
[[1,2,5],[3,4],[6]] => [[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [3,4,1,5,2,6] => 3
[[1,3,4],[2,5],[6]] => [[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [2,5,1,4,3,6] => 3
[[1,2,4],[3,5],[6]] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [3,5,1,4,2,6] => 3
[[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [4,5,1,3,2,6] => 3
[[1,5,6],[2],[3],[4]] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [2,3,4,6,5,1] => 1
[[1,4,6],[2],[3],[5]] => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [2,3,5,6,4,1] => 1
[[1,3,6],[2],[4],[5]] => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [2,4,5,6,3,1] => 1
[[1,2,6],[3],[4],[5]] => [[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [3,4,5,6,2,1] => 1
[[1,4,5],[2],[3],[6]] => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [2,3,5,4,1,6] => 2
[[1,3,5],[2],[4],[6]] => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [2,4,5,3,1,6] => 2
[[1,2,5],[3],[4],[6]] => [[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [3,4,5,2,1,6] => 2
[[1,3,4],[2],[5],[6]] => [[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [2,4,3,1,5,6] => 2
[[1,2,4],[3],[5],[6]] => [[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [3,4,2,1,5,6] => 2
[[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1
[[1,4],[2,5],[3,6]] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [2,3,6,1,4,5] => 2
[[1,3],[2,5],[4,6]] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [2,4,6,1,3,5] => 2
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Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
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
conjugate
Description
Sends a standard tableau to its conjugate tableau.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
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