Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000991: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 1
[1,0,1,0] => [1,2] => [1,2] => 2
[1,1,0,0] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0] => [3,1,2] => [2,3,1] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,4,1,2] => 2
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [2,4,3,1] => 1
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4,3,5,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,3,4,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,5,2,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,3,5,4,2] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [4,2,3,5,1] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [3,2,4,1,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [3,2,5,4,1] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [3,2,4,5,1] => 1
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [2,3,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [3,4,1,2,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [3,5,1,4,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [3,4,1,5,2] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [2,4,3,1,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [4,5,3,1,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [2,5,3,4,1] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [2,4,3,5,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [2,4,5,1,3] => 2
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [4,3,5,1,2] => 2
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [2,3,5,4,1] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [1,2,4,5,3,6] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,5,6,3,4] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,3,2,5,6,4] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,5,3,4,6,2] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,4,3,5,2,6] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,5,3,6,2,4] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,4,3,6,5,2] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,4,3,5,6,2] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,3,4,2,5,6] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,3,4,2,6,5] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,4,6,2,5,3] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,4,5,2,6,3] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,3,5,4,2,6] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,5,6,4,2,3] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,3,6,4,5,2] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,3,5,4,6,2] => 2
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Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see St000314The number of left-to-right-maxima of a permutation..
For the number of left-to-right maxima, see St000314The number of left-to-right-maxima of a permutation..
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
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