Processing math: 100%

Your data matches 64 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000377
Mp00108: Permutations cycle typeInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 0
[1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,1,1]
=> [2,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> 1
[2,1,3] => [2,1]
=> [3]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> 2
[3,2,1] => [2,1]
=> [3]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [3,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [2,2]
=> 1
[1,3,2,4] => [2,1,1]
=> [2,2]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [2,2]
=> 1
[2,1,3,4] => [2,1,1]
=> [2,2]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[3,2,1,4] => [2,1,1]
=> [2,2]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> [4]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 3
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [2,2]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 3
[4,3,2,1] => [2,2]
=> [4]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [4,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [4,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [4,1]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [4,1]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [2,2,1]
=> 2
Description
The dinv defect of an integer partition. This is the number of cells c in the diagram of an integer partition λ for which arm(c)leg(c){0,1}.
Matching statistic: St001176
Mp00108: Permutations cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 0
[1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,1,1]
=> [3]
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> 2
[3,2,1] => [2,1]
=> [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> [2,2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 3
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 3
[4,3,2,1] => [2,2]
=> [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 2
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
Mp00108: Permutations cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 99%distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [1]
=> []
=> 0
[1,2] => [1,1]
=> [2]
=> []
=> 0
[2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,3] => [1,1,1]
=> [3]
=> []
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,6,8,10,11,12,1,2,4,5,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,7,9,10,11,12,1,2,4,5,6,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,5,6,8,11,12,1,2,3,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,5,6,9,10,12,1,2,3,7,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,5,7,8,10,12,1,2,3,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,5,8,9,10,12,1,2,3,6,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,5,8,10,11,12,1,2,3,6,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,6,7,8,11,12,1,2,3,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,6,7,10,11,12,1,2,3,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,6,8,9,11,12,1,2,3,5,7,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,7,8,10,11,12,1,2,3,5,6,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[5,6,7,8,10,12,1,2,3,4,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[5,6,7,9,11,12,1,2,3,4,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[5,6,8,9,10,12,1,2,3,4,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[5,7,8,9,11,12,1,2,3,4,6,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[6,7,8,9,10,12,1,2,3,4,5,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,3,4,5,6,7,8,9,10,12,1,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,1,4,5,6,7,8,9,10,11,12,2] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,5,2,7,4,9,6,11,8,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,5,2,8,4,10,11,7,12,9,6,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,6,2,8,9,5,11,7,4,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,6,2,9,10,5,11,12,8,7,4,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,7,2,8,10,11,6,5,12,9,4,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,6,7,3,9,5,2,11,8,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,6,8,3,10,5,11,7,12,9,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[4,7,8,3,9,11,6,5,2,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[5,6,8,9,4,3,11,7,2,12,10,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[5,6,9,10,4,3,11,12,8,7,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[5,8,9,10,4,11,12,7,6,3,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[6,7,9,10,11,5,4,12,8,3,2,1] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,10,12,6,7,11,9,4,8,5,1,2] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,10,12,8,11,4,5,9,7,6,1,2] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00108: Permutations cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St000394: Dyck paths ⟶ ℤResult quality: 92% values known / values provided: 94%distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 9
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 8
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 9
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 8
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 9
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 8
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,5,7,9,10,12,1,2,4,6,8,11] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[3,5,8,9,11,12,1,2,4,6,7,10] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,6,7,8,10,12,1,2,4,5,9,11] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[3,6,7,9,10,12,1,2,4,5,8,11] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 8
[3,6,7,9,11,12,1,2,4,5,8,10] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[3,6,8,9,10,12,1,2,4,5,7,11] => [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 9
[3,6,8,10,11,12,1,2,4,5,7,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[3,7,8,9,10,12,1,2,4,5,6,11] => [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 8
[3,7,9,10,11,12,1,2,4,5,6,8] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,5,6,8,11,12,1,2,3,7,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,5,6,9,10,12,1,2,3,7,8,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,5,7,8,10,12,1,2,3,6,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,5,8,9,10,12,1,2,3,6,7,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,5,8,10,11,12,1,2,3,6,7,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,6,7,8,11,12,1,2,3,5,9,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,6,7,10,11,12,1,2,3,5,8,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,6,8,9,11,12,1,2,3,5,7,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,7,8,9,10,12,1,2,3,5,6,11] => [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 9
[4,7,8,10,11,12,1,2,3,5,6,9] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[4,8,9,10,11,12,1,2,3,5,6,7] => [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 9
[5,6,7,8,10,12,1,2,3,4,9,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[5,6,7,9,11,12,1,2,3,4,8,10] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[5,6,8,9,10,12,1,2,3,4,7,11] => [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000293
Mp00108: Permutations cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00136: Binary words rotate back-to-frontBinary words
St000293: Binary words ⟶ ℤResult quality: 92% values known / values provided: 94%distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> 10 => 01 => 0
[1,2] => [1,1]
=> 110 => 011 => 0
[2,1] => [2]
=> 100 => 010 => 1
[1,2,3] => [1,1,1]
=> 1110 => 0111 => 0
[1,3,2] => [2,1]
=> 1010 => 0101 => 1
[2,1,3] => [2,1]
=> 1010 => 0101 => 1
[2,3,1] => [3]
=> 1000 => 0100 => 2
[3,1,2] => [3]
=> 1000 => 0100 => 2
[3,2,1] => [2,1]
=> 1010 => 0101 => 1
[1,2,3,4] => [1,1,1,1]
=> 11110 => 01111 => 0
[1,2,4,3] => [2,1,1]
=> 10110 => 01011 => 1
[1,3,2,4] => [2,1,1]
=> 10110 => 01011 => 1
[1,3,4,2] => [3,1]
=> 10010 => 01001 => 2
[1,4,2,3] => [3,1]
=> 10010 => 01001 => 2
[1,4,3,2] => [2,1,1]
=> 10110 => 01011 => 1
[2,1,3,4] => [2,1,1]
=> 10110 => 01011 => 1
[2,1,4,3] => [2,2]
=> 1100 => 0110 => 2
[2,3,1,4] => [3,1]
=> 10010 => 01001 => 2
[2,3,4,1] => [4]
=> 10000 => 01000 => 3
[2,4,1,3] => [4]
=> 10000 => 01000 => 3
[2,4,3,1] => [3,1]
=> 10010 => 01001 => 2
[3,1,2,4] => [3,1]
=> 10010 => 01001 => 2
[3,1,4,2] => [4]
=> 10000 => 01000 => 3
[3,2,1,4] => [2,1,1]
=> 10110 => 01011 => 1
[3,2,4,1] => [3,1]
=> 10010 => 01001 => 2
[3,4,1,2] => [2,2]
=> 1100 => 0110 => 2
[3,4,2,1] => [4]
=> 10000 => 01000 => 3
[4,1,2,3] => [4]
=> 10000 => 01000 => 3
[4,1,3,2] => [3,1]
=> 10010 => 01001 => 2
[4,2,1,3] => [3,1]
=> 10010 => 01001 => 2
[4,2,3,1] => [2,1,1]
=> 10110 => 01011 => 1
[4,3,1,2] => [4]
=> 10000 => 01000 => 3
[4,3,2,1] => [2,2]
=> 1100 => 0110 => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => 011111 => 0
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => 010111 => 1
[1,2,4,3,5] => [2,1,1,1]
=> 101110 => 010111 => 1
[1,2,4,5,3] => [3,1,1]
=> 100110 => 010011 => 2
[1,2,5,3,4] => [3,1,1]
=> 100110 => 010011 => 2
[1,2,5,4,3] => [2,1,1,1]
=> 101110 => 010111 => 1
[1,3,2,4,5] => [2,1,1,1]
=> 101110 => 010111 => 1
[1,3,2,5,4] => [2,2,1]
=> 11010 => 01101 => 2
[1,3,4,2,5] => [3,1,1]
=> 100110 => 010011 => 2
[1,3,4,5,2] => [4,1]
=> 100010 => 010001 => 3
[1,3,5,2,4] => [4,1]
=> 100010 => 010001 => 3
[1,3,5,4,2] => [3,1,1]
=> 100110 => 010011 => 2
[1,4,2,3,5] => [3,1,1]
=> 100110 => 010011 => 2
[1,4,2,5,3] => [4,1]
=> 100010 => 010001 => 3
[1,4,3,2,5] => [2,1,1,1]
=> 101110 => 010111 => 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => 010011 => 2
[1,4,5,2,3] => [2,2,1]
=> 11010 => 01101 => 2
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> 100000000000 => ? => ? = 10
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> 100000000000 => ? => ? = 10
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> 100000000000 => ? => ? = 10
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> 10000000100 => 01000000010 => ? = 9
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> 100000000000 => ? => ? = 10
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> 100000000000 => ? => ? = 10
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> 100000000000 => ? => ? = 10
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> 100000000000 => ? => ? = 10
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> 1000000000000 => ? => ? = 11
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> 10000001000 => ? => ? = 10
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> 1000000000000 => ? => ? = 11
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> 1000000000000 => ? => ? = 11
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> 10000001000 => ? => ? = 10
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> 1000000000000 => ? => ? = 11
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> 1000010100 => ? => ? = 9
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> 1000000000000 => ? => ? = 11
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> 1000000000000 => ? => ? = 11
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> 1000000000000 => ? => ? = 11
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> 1000000000000 => ? => ? = 11
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> 10000001000 => ? => ? = 10
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> 1000010100 => ? => ? = 9
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> 10000001100 => ? => ? = 9
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> 1000000000000 => ? => ? = 11
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> 10000001100 => ? => ? = 9
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> 1000000000000 => ? => ? = 11
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> 1000011100 => 0100001110 => ? = 8
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> 10000001100 => ? => ? = 9
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> 1000000000000 => ? => ? = 11
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> 1000010100 => ? => ? = 9
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> 10000001000 => ? => ? = 10
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> 1000000000000 => ? => ? = 11
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> 10000001000 => ? => ? = 10
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> 1000000000000 => ? => ? = 11
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> 1000000000000 => ? => ? = 11
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> 10000001000 => ? => ? = 10
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> 10000001000 => ? => ? = 10
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> 1000000000000 => ? => ? = 11
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> 10000001000 => ? => ? = 10
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> 1000000000000 => ? => ? = 11
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> 1000000000000 => ? => ? = 11
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> 100000000100 => 010000000010 => ? = 10
Description
The number of inversions of a binary word.
Mp00108: Permutations cycle typeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 94% values known / values provided: 94%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[2,1] => [2]
=> [[1,2]]
=> 2 = 1 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[3,1,2] => [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 + 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 9 + 1
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? = 9 + 1
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? = 9 + 1
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 9 + 1
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 5 + 1
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 5 + 1
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 10 + 1
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 10 + 1
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 9 + 1
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? = 9 + 1
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 10 + 1
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 9 + 1
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? = 9 + 1
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 10 + 1
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 9 + 1
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 10 + 1
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[2,7,8,9,10,12,1,3,4,5,6,11] => [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 7 + 1
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? = 9 + 1
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
[3,4,6,8,11,12,1,2,5,7,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 9 + 1
[3,4,7,8,10,12,1,2,5,6,9,11] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 9 + 1
[3,4,7,8,11,12,1,2,5,6,9,10] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 8 + 1
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
[3,4,7,9,11,12,1,2,5,6,8,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 9 + 1
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? = 10 + 1
Description
The number of ascents of a standard tableau. Entry i of a standard Young tableau is an '''ascent''' if i+1 appears to the right or above i in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000738
Mp00108: Permutations cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 92% values known / values provided: 93%distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,1] => [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,3] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[3,1,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[3,2,1] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ? = 9 + 1
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 9 + 1
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 9 + 1
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 9 + 1
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 5 + 1
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 5 + 1
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 + 1
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ? = 10 + 1
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [3,3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 9 + 1
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ? = 10 + 1
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [3,3,2,1,1,1,1]
=> [[1,6,9],[2,8,12],[3,11],[4],[5],[7],[10]]
=> ? = 9 + 1
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 9 + 1
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ? = 10 + 1
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [3,3,2,1,1,1,1]
=> [[1,6,9],[2,8,12],[3,11],[4],[5],[7],[10]]
=> ? = 9 + 1
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 9 + 1
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Mp00108: Permutations cycle typeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 0
[1,2] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[2,1] => [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[3,1,2] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 3
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 2
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 9
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 9
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10],[11]]
=> ? = 9
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 5
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 5
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ?
=> ? = 10
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 10
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 10
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ?
=> ? = 9
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ?
=> ? = 10
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ?
=> ? = 9
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 10
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> [[1,3,8],[2,4,9],[5,10],[6,11],[7,12]]
=> ? = 9
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ?
=> ? = 10
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ?
=> ? = 9
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 10
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 9
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 8
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> [[1,3,8],[2,4,9],[5,10],[6,11],[7,12]]
=> ? = 9
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 10
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 9
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 8
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 9
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 10
Description
The number of descents of a standard tableau. Entry i of a standard Young tableau is a descent if i+1 appears in a row below the row of i.
Mp00108: Permutations cycle typeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000245: Permutations ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? = 9
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 9
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 9
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? = 9
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 10
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 9
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 10
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 9
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 9
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 10
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 9
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 9
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 8
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 9
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 9
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
Description
The number of ascents of a permutation.
Matching statistic: St000441
Mp00108: Permutations cycle typeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000441: Permutations ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? = 9
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 9
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 9
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? = 9
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 5
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 10
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 9
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 10
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 9
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 9
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 10
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 9
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 9
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 8
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 9
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 10
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 9
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 10
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
Description
The number of successions of a permutation. A succession of a permutation π is an index i such that π(i)+1=π(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000672The number of minimal elements in Bruhat order not less than the permutation. St000074The number of special entries. St000369The dinv deficit of a Dyck path. St000676The number of odd rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000502The number of successions of a set partitions. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000211The rank of the set partition. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000702The number of weak deficiencies of a permutation. St000332The positive inversions of an alternating sign matrix. St000740The last entry of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000316The number of non-left-to-right-maxima of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001480The number of simple summands of the module J^2/J^3. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St000213The number of weak exceedances (also weak excedences) of a permutation. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.