Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000996: Permutations ⟶ ℤ
Values
[1] => [1,0] => [1] => [1] => 0
[2] => [1,0,1,0] => [1,2] => [1,2] => 0
[1,1] => [1,1,0,0] => [2,1] => [1,2] => 0
[3] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 1
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => [1,2,3] => 0
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 1
[2,2] => [1,1,1,0,0,0] => [3,1,2] => [1,2,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,4,2] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,2,3] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => [1,4,2,3] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,3,4,5,2] => 3
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => [1,2,3,4] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,4,2,5,3] => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => 3
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => [1,2,3,4] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [1,4,5,2,3] => 2
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => 4
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 1
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => 2
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,5,3,6,4] => 2
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => 3
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [1,5,2,3,4] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,4,2,5,6,3] => 3
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => 4
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [1,2,5,3,4] => 1
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => [1,4,5,6,2,3] => 3
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => 5
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => 1
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => 2
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => 2
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [1,2,3,4,5] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,4,5,2,6,3] => 3
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 1
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => 3
[4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,8,4] => 4
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [1,2,4,3,5] => 1
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => [1,5,6,2,3,4] => 2
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,6,4] => [1,5,2,3,6,4] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,6,7,3] => [1,4,2,5,6,7,3] => 4
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [1,2,3,4,5] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => [1,2,5,6,3,4] => 2
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [3,1,4,5,6,7,2] => [1,4,5,6,7,2,3] => 4
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => 0
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => 1
[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => 1
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => 2
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,4,5,6,2,3] => 3
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => 3
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => 1
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => [1,6,2,3,4,5] => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,4,6,2,3,5] => 2
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,2,6,7,3] => [1,4,5,2,6,7,3] => 4
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [1,2,3,4,5] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => [1,2,6,3,5,4] => 1
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [3,4,1,5,6,7,2] => [1,5,6,7,2,3,4] => 3
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,2,3,4] => [1,5,6,2,3,4] => 2
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => [1,2,6,3,4,5] => 1
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [4,1,2,5,6,7,3] => [1,2,5,6,7,3,4] => 3
[1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => 0
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[9,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => 1
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => 3
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => [1,2,3,4,5,6] => 0
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,2,7,3] => [1,4,5,6,2,7,3] => 4
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => 2
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => [1,2,5,3,4,6] => 1
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [3,4,5,1,6,7,2] => [1,6,7,2,3,4,5] => 2
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 1
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,6,2,3,7,5] => [1,4,6,2,3,7,5] => 3
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,3,4,5] => [1,2,6,7,3,4,5] => 2
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => [1,2,3,6,4,5] => 1
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => [1,2,4,3,5,6] => 1
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [3,5,1,2,6,7,4] => [1,2,6,7,3,5,4] => 2
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [4,5,1,2,6,7,3] => [1,2,6,7,3,4,5] => 2
[1,1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => 0
[10,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => 1
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,2,3] => [1,4,5,6,7,2,3] => 4
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [3,4,5,6,1,7,2] => [1,7,2,3,4,5,6] => 1
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,2,3,6] => [1,4,5,7,2,3,6] => 3
>>> Load all 167 entries. <<<
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searching the database for statistics with the same generating function
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
This is the number of left-to-right maxima that are not right-to-left minima.
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.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
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