Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000141: Permutations ⟶ ℤ
Values
[1] => [1,0,1,0] => [2,1] => [2,1] => 1
[2] => [1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 1
[1,1] => [1,0,1,1,0,0] => [2,3,1] => [2,1,3] => 1
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,2,4,3] => 1
[2,1] => [1,0,1,0,1,0] => [3,2,1] => [3,2,1] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,1,3,4] => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,4,3,1] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,1,3,4,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,3,5,4,1] => 2
[3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,4,3,2] => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => [4,2,3,1] => 3
[2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,2,1,4] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [3,2,4,5,1] => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => [2,3,4,6,5,1] => 2
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,4,2] => 2
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,5,3,4,1] => 3
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,4,5,2,3] => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => 3
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [4,2,3,5,1] => 3
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [3,4,1,2,5] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [3,2,4,1,5] => 2
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => [3,2,4,5,6,1] => 2
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 1
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => [1,3,4,6,5,2] => 2
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => [2,3,6,4,5,1] => 3
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,2,5,4,3] => 2
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,5,4,2,1] => 3
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [5,2,3,4,1] => 4
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [4,5,2,1,3] => 3
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [4,5,1,3,2] => 3
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [4,3,5,2,1] => 3
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => [4,2,3,5,6,1] => 3
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [3,2,1,4,5] => 2
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [3,2,4,5,1,6] => 2
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => 2
[1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [2,1,3,4,5,6,7,8] => 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 1
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,4,6,5,3] => 2
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => [3,4,6,5,2,1] => 3
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => [2,6,3,4,5,1] => 4
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [2,5,4,3,1] => 3
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,5,1,4,2] => 3
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [5,3,4,2,1] => 4
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => [5,2,3,4,6,1] => 4
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [4,5,3,1,2] => 3
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [4,2,5,1,3] => 3
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [4,3,2,5,1] => 3
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [4,3,5,6,2,1] => 3
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [4,2,3,5,6,7,1] => [4,2,3,5,6,7,1] => 3
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [3,4,1,2,5,6] => 2
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [3,2,4,1,5,6] => 2
[2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,8,1] => 2
[1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [2,1,3,4,5,6,7,8,9] => 1
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 1
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 2
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => [2,4,6,5,3,1] => 3
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,4,6,1,5,2] => 3
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => [3,6,4,5,2,1] => 4
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => 5
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => [2,5,6,3,1,4] => 3
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,5,4,3,2] => 3
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [5,2,4,3,1] => 4
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [5,3,2,4,1] => 4
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [5,3,4,6,2,1] => 4
[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [5,2,3,4,6,7,1] => [5,2,3,4,6,7,1] => 4
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => 3
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,3,2,1,5] => 3
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [4,2,5,6,1,3] => 3
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => [4,5,1,3,6,2] => 3
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => 3
[3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,8,1] => 3
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [3,2,1,4,5,6] => 2
[2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [2,1,3,4,5,6,7,8,9,10] => 1
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => 1
[7,1,1,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [8,2,3,4,1,5,6,7] => [2,3,4,8,5,6,7,1] => 4
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => [2,3,6,5,4,1] => 3
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => [1,4,6,5,3,2] => 3
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => [2,6,4,5,3,1] => 4
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => [3,6,4,2,5,1] => 4
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [6,3,4,5,2,1] => 5
[5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [6,2,3,4,5,7,1] => [6,2,3,4,5,7,1] => 5
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => [1,5,6,3,2,4] => 3
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => [2,5,3,6,1,4] => 3
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => [1,5,6,2,4,3] => 3
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [5,2,4,6,3,1] => 4
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => [3,5,1,4,6,2] => 3
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [5,3,4,2,6,1] => 4
[4,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,8,1] => 4
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => [4,5,2,1,3,6] => 3
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => [4,5,1,3,2,6] => 3
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [4,3,5,2,1,6] => 3
>>> Load all 230 entries. <<<
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searching the database for the individual values of this statistic
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search for generating function
searching the database for statistics with the same generating function
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
Map
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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