Your data matches 55 different statistics following compositions of up to 3 maps.
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Mp00204: Permutations LLPSInteger partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [1,1]
=> 1
[2,1] => [2]
=> 0
[1,2,3] => [1,1,1]
=> 2
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [2,1]
=> 1
[3,1,2] => [2,1]
=> 1
[3,2,1] => [3]
=> 0
[1,2,3,4] => [1,1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> 2
[1,4,3,2] => [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [2,1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> 2
[2,4,3,1] => [3,1]
=> 1
[3,1,2,4] => [2,1,1]
=> 2
[3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [3,1]
=> 1
[3,2,4,1] => [3,1]
=> 1
[3,4,1,2] => [2,1,1]
=> 2
[3,4,2,1] => [3,1]
=> 1
[4,1,2,3] => [2,1,1]
=> 2
[4,1,3,2] => [3,1]
=> 1
[4,2,1,3] => [3,1]
=> 1
[4,2,3,1] => [3,1]
=> 1
[4,3,1,2] => [3,1]
=> 1
[4,3,2,1] => [4]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> 3
[1,4,3,2,5] => [3,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> 3
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.
Mp00204: Permutations LLPSInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1] => [3]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> 1
[4,3,2,1] => [4]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00204: Permutations LLPSInteger partitions
Mp00044: Integer partitions conjugateInteger 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]
=> [1]
=> 0
[1,2] => [1,1]
=> [2]
=> [1,1]
=> 1
[2,1] => [2]
=> [1,1]
=> [2]
=> 0
[1,2,3] => [1,1,1]
=> [3]
=> [1,1,1]
=> 2
[1,3,2] => [2,1]
=> [2,1]
=> [3]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [3]
=> 1
[2,3,1] => [2,1]
=> [2,1]
=> [3]
=> 1
[3,1,2] => [2,1]
=> [2,1]
=> [3]
=> 1
[3,2,1] => [3]
=> [1,1,1]
=> [2,1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2,2]
=> [4]
=> 2
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2,2]
=> [4]
=> 2
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,3,2,1] => [4]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [5]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [3,2]
=> [5]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001034
Mp00204: Permutations LLPSInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> 0
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 1
[2,1] => [2]
=> []
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 1
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 1
[2,3,1] => [2,1]
=> [1]
=> [1,0]
=> 1
[3,1,2] => [2,1]
=> [1]
=> [1,0]
=> 1
[3,2,1] => [3]
=> []
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,3,2,1] => [4]
=> []
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Mp00204: Permutations LLPSInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 91% values known / values provided: 100%distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> []
=> => ? = 0
[1,2] => [1,1]
=> [1]
=> 10 => 1
[2,1] => [2]
=> []
=> => ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 110 => 2
[1,3,2] => [2,1]
=> [1]
=> 10 => 1
[2,1,3] => [2,1]
=> [1]
=> 10 => 1
[2,3,1] => [2,1]
=> [1]
=> 10 => 1
[3,1,2] => [2,1]
=> [1]
=> 10 => 1
[3,2,1] => [3]
=> []
=> => ? = 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[1,4,3,2] => [3,1]
=> [1]
=> 10 => 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,1,4,3] => [2,2]
=> [2]
=> 100 => 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,4,3,1] => [3,1]
=> [1]
=> 10 => 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[3,1,4,2] => [2,2]
=> [2]
=> 100 => 2
[3,2,1,4] => [3,1]
=> [1]
=> 10 => 1
[3,2,4,1] => [3,1]
=> [1]
=> 10 => 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> 110 => 2
[3,4,2,1] => [3,1]
=> [1]
=> 10 => 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[4,1,3,2] => [3,1]
=> [1]
=> 10 => 1
[4,2,1,3] => [3,1]
=> [1]
=> 10 => 1
[4,2,3,1] => [3,1]
=> [1]
=> 10 => 1
[4,3,1,2] => [3,1]
=> [1]
=> 10 => 1
[4,3,2,1] => [4]
=> []
=> => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1010 => 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 1010 => 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 110 => 2
[5,4,3,2,1] => [5]
=> []
=> => ? = 0
[6,5,4,3,2,1] => [6]
=> []
=> => ? = 0
[7,6,5,4,3,2,1] => [7]
=> []
=> => ? = 0
[8,7,6,5,4,3,2,1] => [8]
=> []
=> => ? = 0
[10,9,8,7,6,5,4,3,2,1] => [10]
=> []
=> => ? = 0
[12,11,10,9,8,7,6,5,4,3,2,1] => [12]
=> []
=> => ? = 0
[9,8,7,6,5,4,3,2,1] => [9]
=> []
=> => ? = 0
[11,10,9,8,7,6,5,4,3,2,1] => [11]
=> []
=> => ? = 0
Description
The number of inversions of a binary word.
Mp00204: Permutations LLPSInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 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] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,3,4,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,4,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,3,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,4,2,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,1,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,2,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,3,1,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 4 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,4,5,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,5,3,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,4,5,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,5,2,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,4,2,5,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,5,2,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,4,3,6,5,10,9,8,7,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,4,3,6,5,12,9,8,11,10,7] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,4,3,6,5,12,11,10,9,8,7] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,4,3,8,7,6,5,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,4,3,10,7,6,9,8,5,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,4,3,12,7,6,9,8,11,10,5] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,4,3,12,7,6,11,10,9,8,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,4,3,10,9,8,7,6,5,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,4,3,12,9,8,7,6,11,10,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,4,3,12,11,8,7,10,9,6,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,4,3,12,11,10,9,8,7,6,5] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[2,1,6,5,4,3,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,8,5,4,7,6,3,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,10,5,4,7,6,9,8,3,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,12,5,4,7,6,9,8,11,10,3] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[2,1,12,5,4,7,6,11,10,9,8,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,10,5,4,9,8,7,6,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,12,5,4,9,8,7,6,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,12,5,4,11,8,7,10,9,6,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,12,5,4,11,10,9,8,7,6,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[2,1,8,7,6,5,4,3,10,9,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,10,7,6,5,4,9,8,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,12,7,6,5,4,9,8,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,10,9,6,5,8,7,4,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,12,9,6,5,8,7,4,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,12,11,6,5,8,7,10,9,4,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[2,1,12,11,6,5,10,9,8,7,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[2,1,10,9,8,7,6,5,4,3,12,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[2,1,12,9,8,7,6,5,4,11,10,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[2,1,12,11,8,7,6,5,10,9,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[2,1,12,11,10,7,6,9,8,5,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[2,1,12,11,10,9,8,7,6,5,4,3] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 2 + 1
[4,3,2,1,6,5,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[4,3,2,1,8,7,6,5,12,11,10,9] => [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 8 + 1
[4,3,2,1,12,11,10,9,8,7,6,5] => [8,4]
=> [[1,2,3,4,9,10,11,12],[5,6,7,8]]
=> ? = 4 + 1
[6,3,2,5,4,1,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[8,3,2,5,4,7,6,1,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[10,3,2,5,4,7,6,9,8,1,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[12,3,2,5,4,7,6,9,8,11,10,1] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 8 + 1
[12,3,2,5,4,7,6,11,10,9,8,1] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[10,3,2,5,4,9,8,7,6,1,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[12,3,2,5,4,9,8,7,6,11,10,1] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[12,3,2,5,4,11,8,7,10,9,6,1] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[12,3,2,5,4,11,10,9,8,7,6,1] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 4 + 1
[8,3,2,7,6,5,4,1,10,9,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[10,3,2,7,6,5,4,9,8,1,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[12,3,2,7,6,5,4,9,8,11,10,1] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[10,3,2,9,6,5,8,7,4,1,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
[12,3,2,9,6,5,8,7,4,11,10,1] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 6 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Mp00204: Permutations LLPSInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[2,1] => [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,3,2] => [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,2,1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,1,4,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,2,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,4,2,5,3] => [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,4,3,6,5,10,9,8,7,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,4,3,6,5,12,9,8,11,10,7] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,4,3,6,5,12,11,10,9,8,7] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,4,3,8,7,6,5,10,9,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,4,3,10,7,6,9,8,5,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,4,3,12,7,6,9,8,11,10,5] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,4,3,12,7,6,11,10,9,8,5] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,4,3,10,9,8,7,6,5,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,4,3,12,9,8,7,6,11,10,5] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,4,3,12,11,8,7,10,9,6,5] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,4,3,12,11,10,9,8,7,6,5] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,1,6,5,4,3,8,7,10,9,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,8,5,4,7,6,3,10,9,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,10,5,4,7,6,9,8,3,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,12,5,4,7,6,9,8,11,10,3] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[2,1,12,5,4,7,6,11,10,9,8,3] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,10,5,4,9,8,7,6,3,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,12,5,4,9,8,7,6,11,10,3] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,12,5,4,11,8,7,10,9,6,3] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,12,5,4,11,10,9,8,7,6,3] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,1,8,7,6,5,4,3,10,9,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,10,7,6,5,4,9,8,3,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,12,7,6,5,4,9,8,11,10,3] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,10,9,6,5,8,7,4,3,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,12,9,6,5,8,7,4,11,10,3] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,12,11,6,5,8,7,10,9,4,3] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[2,1,12,11,6,5,10,9,8,7,4,3] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,1,10,9,8,7,6,5,4,3,12,11] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,1,12,9,8,7,6,5,4,11,10,3] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,1,12,11,8,7,6,5,10,9,4,3] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,1,12,11,10,7,6,9,8,5,4,3] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[2,1,12,11,10,9,8,7,6,5,4,3] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2 + 1
[4,3,2,1,6,5,8,7,10,9,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[4,3,2,1,8,7,6,5,12,11,10,9] => [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 8 + 1
[4,3,2,1,12,11,10,9,8,7,6,5] => [8,4]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 4 + 1
[6,3,2,5,4,1,8,7,10,9,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[8,3,2,5,4,7,6,1,10,9,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[10,3,2,5,4,7,6,9,8,1,12,11] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[12,3,2,5,4,7,6,9,8,11,10,1] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 8 + 1
[12,3,2,5,4,7,6,11,10,9,8,1] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[10,3,2,5,4,9,8,7,6,1,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[12,3,2,5,4,9,8,7,6,11,10,1] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[12,3,2,5,4,11,8,7,10,9,6,1] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[12,3,2,5,4,11,10,9,8,7,6,1] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 4 + 1
[8,3,2,7,6,5,4,1,10,9,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[10,3,2,7,6,5,4,9,8,1,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[12,3,2,7,6,5,4,9,8,11,10,1] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[10,3,2,9,6,5,8,7,4,1,12,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 1
[12,3,2,9,6,5,8,7,4,11,10,1] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 6 + 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: St000734
Mp00204: Permutations LLPSInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 90% values known / values provided: 90%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 1 + 1
[2,1] => [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,2,1] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,3,4,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3 = 2 + 1
[2,3,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,3,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3 = 2 + 1
[3,2,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,4,2,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,1,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,1,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,4,5,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,3,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,4,5,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,2,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,2,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,4,2,5,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,4,5,2,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,4,3,6,5,10,9,8,7,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,4,3,6,5,12,9,8,11,10,7] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,4,3,6,5,12,11,10,9,8,7] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,4,3,8,7,6,5,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,4,3,8,7,6,5,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[2,1,4,3,10,7,6,9,8,5,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,4,3,12,7,6,9,8,11,10,5] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,4,3,12,7,6,11,10,9,8,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,4,3,10,9,8,7,6,5,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,4,3,12,9,8,7,6,11,10,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,4,3,12,11,8,7,10,9,6,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,4,3,12,11,10,9,8,7,6,5] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 4 + 1
[2,1,6,5,4,3,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,6,5,4,3,8,7,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[2,1,6,5,4,3,10,9,8,7,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[2,1,6,5,4,3,12,9,8,11,10,7] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[2,1,6,5,4,3,12,11,10,9,8,7] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
=> ? = 6 + 1
[2,1,8,5,4,7,6,3,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,8,5,4,7,6,3,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[2,1,10,5,4,7,6,9,8,3,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,12,5,4,7,6,9,8,11,10,3] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[2,1,12,5,4,7,6,11,10,9,8,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,10,5,4,9,8,7,6,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,5,4,9,8,7,6,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,5,4,11,8,7,10,9,6,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,5,4,11,10,9,8,7,6,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 4 + 1
[2,1,8,7,6,5,4,3,10,9,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,8,7,6,5,4,3,12,11,10,9] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
=> ? = 6 + 1
[2,1,10,7,6,5,4,9,8,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,7,6,5,4,9,8,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,7,6,5,4,11,10,9,8,3] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
=> ? = 6 + 1
[2,1,10,9,6,5,8,7,4,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,9,6,5,8,7,4,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,11,6,5,8,7,10,9,4,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 6 + 1
[2,1,12,11,6,5,10,9,8,7,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 4 + 1
[2,1,10,9,8,7,6,5,4,3,12,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 4 + 1
[2,1,12,9,8,7,6,5,4,11,10,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 4 + 1
[2,1,12,11,8,7,6,5,10,9,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 4 + 1
[2,1,12,11,10,7,6,9,8,5,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 4 + 1
[2,1,12,11,10,9,8,7,6,5,4,3] => [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]]
=> ? = 2 + 1
[4,3,2,1,6,5,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 8 + 1
[4,3,2,1,6,5,8,7,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[4,3,2,1,6,5,10,9,8,7,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[4,3,2,1,6,5,12,9,8,11,10,7] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[4,3,2,1,6,5,12,11,10,9,8,7] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
=> ? = 6 + 1
[4,3,2,1,8,7,6,5,10,9,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[4,3,2,1,8,7,6,5,12,11,10,9] => [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8,12]]
=> ? = 8 + 1
[4,3,2,1,10,7,6,9,8,5,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
[4,3,2,1,12,7,6,9,8,11,10,5] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 8 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000141
Mp00204: Permutations LLPSInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000141: Permutations ⟶ ℤResult quality: 90% values known / values provided: 90%distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,2,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[1,3,4,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[1,4,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[1,4,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,3,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,4,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[3,1,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[3,4,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,4,2,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,1,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,2,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,3,1,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,2,4,5,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,2,5,3,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 3
[1,3,4,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,3,4,5,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,3,5,2,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,2,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[1,4,2,5,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 3
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,5,2,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3
[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] => ? = 10
[2,1,4,3,6,5,8,7,12,11,10,9] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,4,3,6,5,10,9,8,7,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,4,3,6,5,12,9,8,11,10,7] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,4,3,6,5,12,11,10,9,8,7] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,4,3,8,7,6,5,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[2,1,4,3,10,7,6,9,8,5,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,4,3,12,7,6,9,8,11,10,5] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,4,3,12,7,6,11,10,9,8,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 4
[2,1,6,5,4,3,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,6,5,4,3,8,7,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[2,1,6,5,4,3,10,9,8,7,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[2,1,6,5,4,3,12,9,8,11,10,7] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[2,1,6,5,4,3,12,11,10,9,8,7] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 6
[2,1,8,5,4,7,6,3,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,8,5,4,7,6,3,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[2,1,10,5,4,7,6,9,8,3,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,12,5,4,7,6,9,8,11,10,3] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[2,1,12,5,4,7,6,11,10,9,8,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,10,5,4,9,8,7,6,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,5,4,9,8,7,6,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,5,4,11,8,7,10,9,6,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,5,4,11,10,9,8,7,6,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 4
[2,1,8,7,6,5,4,3,10,9,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,8,7,6,5,4,3,12,11,10,9] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 6
[2,1,10,7,6,5,4,9,8,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,7,6,5,4,9,8,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,7,6,5,4,11,10,9,8,3] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 6
[2,1,10,9,6,5,8,7,4,3,12,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,9,6,5,8,7,4,11,10,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,11,6,5,8,7,10,9,4,3] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 6
[2,1,12,11,6,5,10,9,8,7,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 4
[2,1,10,9,8,7,6,5,4,3,12,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 4
[2,1,12,9,8,7,6,5,4,11,10,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 4
[2,1,12,11,8,7,6,5,10,9,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 4
[2,1,12,11,10,7,6,9,8,5,4,3] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 4
[2,1,12,11,10,9,8,7,6,5,4,3] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[4,3,2,1,6,5,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 8
[4,3,2,1,6,5,8,7,12,11,10,9] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[4,3,2,1,6,5,10,9,8,7,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[4,3,2,1,6,5,12,9,8,11,10,7] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[4,3,2,1,6,5,12,11,10,9,8,7] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 6
[4,3,2,1,8,7,6,5,10,9,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
[4,3,2,1,8,7,6,5,12,11,10,9] => [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? = 8
[4,3,2,1,10,7,6,9,8,5,12,11] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 8
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 $i-\pi(i)$.
Matching statistic: St000662
Mp00204: Permutations LLPSInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000662: Permutations ⟶ ℤResult quality: 90% values known / values provided: 90%distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[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] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,2,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[2,3,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[2,4,1,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,4,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[3,4,2,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,1,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,3,1,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,2,4,5,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,2,5,3,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3
[1,3,4,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,3,4,5,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,3,5,2,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 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] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[1,4,2,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3
[1,4,3,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,5,2,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3
[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] => ? = 10
[2,1,4,3,6,5,8,7,12,11,10,9] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,4,3,6,5,10,9,8,7,12,11] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,4,3,6,5,12,9,8,11,10,7] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,4,3,6,5,12,11,10,9,8,7] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,4,3,8,7,6,5,12,11,10,9] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[2,1,4,3,10,7,6,9,8,5,12,11] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,4,3,12,7,6,9,8,11,10,5] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,4,3,12,7,6,11,10,9,8,5] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,1,6,5,4,3,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,6,5,4,3,8,7,12,11,10,9] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[2,1,6,5,4,3,10,9,8,7,12,11] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[2,1,6,5,4,3,12,9,8,11,10,7] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[2,1,6,5,4,3,12,11,10,9,8,7] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 6
[2,1,8,5,4,7,6,3,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,8,5,4,7,6,3,12,11,10,9] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[2,1,10,5,4,7,6,9,8,3,12,11] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,12,5,4,7,6,9,8,11,10,3] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[2,1,12,5,4,7,6,11,10,9,8,3] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,10,5,4,9,8,7,6,3,12,11] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,5,4,9,8,7,6,11,10,3] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,5,4,11,8,7,10,9,6,3] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,5,4,11,10,9,8,7,6,3] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,1,8,7,6,5,4,3,10,9,12,11] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,8,7,6,5,4,3,12,11,10,9] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 6
[2,1,10,7,6,5,4,9,8,3,12,11] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,7,6,5,4,9,8,11,10,3] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,7,6,5,4,11,10,9,8,3] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 6
[2,1,10,9,6,5,8,7,4,3,12,11] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,9,6,5,8,7,4,11,10,3] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,11,6,5,8,7,10,9,4,3] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 6
[2,1,12,11,6,5,10,9,8,7,4,3] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,1,10,9,8,7,6,5,4,3,12,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,1,12,9,8,7,6,5,4,11,10,3] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,1,12,11,8,7,6,5,10,9,4,3] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,1,12,11,10,7,6,9,8,5,4,3] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,1,12,11,10,9,8,7,6,5,4,3] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 2
[4,3,2,1,6,5,8,7,10,9,12,11] => [4,2,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,1,2,3,4] => ? = 8
[4,3,2,1,6,5,8,7,12,11,10,9] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[4,3,2,1,6,5,10,9,8,7,12,11] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[4,3,2,1,6,5,12,9,8,11,10,7] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[4,3,2,1,6,5,12,11,10,9,8,7] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 6
[4,3,2,1,8,7,6,5,10,9,12,11] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
[4,3,2,1,8,7,6,5,12,11,10,9] => [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? = 8
[4,3,2,1,10,7,6,9,8,5,12,11] => [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 8
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000441The number of successions of a permutation. 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. St000010The length of the partition. St001726The number of visible inversions of a permutation. St000839The largest opener of a set partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000502The number of successions of a set partitions. St000211The rank of the set partition. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000362The size of a minimal vertex cover of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001298The number of repeated entries in the Lehmer code of a permutation. St000470The number of runs in 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. St000354The number of recoils of a permutation. St000209Maximum difference of elements in cycles. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000956The maximal displacement of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. St001668The number of points of the poset minus the width of the poset. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001626The number of maximal proper sublattices of a lattice.