Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000358: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0] => [2,1,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,1,0,0] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0] => [3,2,1] => [1,3,2] => [2,3,1] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,2,4,3] => [2,3,4,1] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 0
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [1,3,2,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [1,3,4,2] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [1,3,2,4] => [2,3,1,4] => 0
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,4,2,3] => [2,1,4,3] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,2,4,3,5] => [2,3,4,1,5] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,2,4,5,3] => [2,3,5,1,4] => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,2,4,3,5] => [2,3,4,1,5] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,2,5,3,4] => [2,3,1,5,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => 0
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,5,1] => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [1,2,4,3,5] => [2,3,4,1,5] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [1,2,4,3,5] => [2,3,4,1,5] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [1,3,2,4,5] => [2,3,1,4,5] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [1,3,4,2,5] => [2,4,1,3,5] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [1,3,4,5,2] => [2,5,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [1,3,5,2,4] => [2,1,4,5,3] => 0
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [1,3,2,4,5] => [2,3,1,4,5] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [1,3,2,4,5] => [2,3,1,4,5] => 0
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [1,3,5,2,4] => [2,1,4,5,3] => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [1,3,4,2,5] => [2,4,1,3,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [1,4,2,3,5] => [2,1,4,3,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [1,4,5,2,3] => [2,1,5,3,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [1,4,2,3,5] => [2,1,4,3,5] => 0
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [1,4,2,5,3] => [3,4,5,1,2] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,5,2,4,3] => [3,2,5,4,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,4,6,5] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,3,4,6,5] => [2,3,4,5,6,1] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,3,5,4,6] => [2,3,4,5,1,6] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,3,5,6,4] => [2,3,4,6,1,5] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,3,5,4,6] => [2,3,4,5,1,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,3,6,4,5] => [2,3,4,1,6,5] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,2,3,4,6,5] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,2,3,4,6,5] => [2,3,4,5,6,1] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,2,3,5,4,6] => [2,3,4,5,1,6] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,2,3,5,6,4] => [2,3,4,6,1,5] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,2,3,5,4,6] => [2,3,4,5,1,6] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,2,3,6,4,5] => [2,3,4,1,6,5] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,2,4,5,3,6] => [2,3,5,1,4,6] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,2,4,5,6,3] => [2,3,6,1,4,5] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,2,4,6,3,5] => [2,3,1,5,6,4] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,2,4,6,3,5] => [2,3,1,5,6,4] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,2,4,5,3,6] => [2,3,5,1,4,6] => 1
>>> Load all 260 entries. <<<
search for individual values
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 number of occurrences of the pattern 31-2.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern $31\!\!-\!\!2$.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern $31\!\!-\!\!2$.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to 312-avoiding permutation
Description
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
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