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Your data matches 64 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [2]
=> 0
[2,1] => [1,1]
=> 1
[1,2,3] => [3]
=> 0
[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] => [1,1,1]
=> 2
[1,2,3,4] => [4]
=> 0
[1,2,4,3] => [3,1]
=> 1
[1,3,2,4] => [3,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [3,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [3,1]
=> 1
[2,4,1,3] => [2,2]
=> 2
[2,4,3,1] => [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [2,1,1]
=> 2
[4,1,2,3] => [3,1]
=> 1
[4,1,3,2] => [2,1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> 3
[1,2,3,4,5] => [5]
=> 0
[1,2,3,5,4] => [4,1]
=> 1
[1,2,4,3,5] => [4,1]
=> 1
[1,2,4,5,3] => [4,1]
=> 1
[1,2,5,3,4] => [4,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> 2
[1,3,2,4,5] => [4,1]
=> 1
[1,3,2,5,4] => [3,2]
=> 2
[1,3,4,2,5] => [4,1]
=> 1
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [3,2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> 2
[1,4,2,3,5] => [4,1]
=> 1
[1,4,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [3,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> 2
[1,4,5,2,3] => [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
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer 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] => [2]
=> []
=> 0
[2,1] => [1,1]
=> [1]
=> 1
[1,2,3] => [3]
=> []
=> 0
[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] => [1,1,1]
=> [1,1]
=> 2
[1,2,3,4] => [4]
=> []
=> 0
[1,2,4,3] => [3,1]
=> [1]
=> 1
[1,3,2,4] => [3,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> 1
[1,4,2,3] => [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 2
[2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> 1
[2,3,4,1] => [3,1]
=> [1]
=> 1
[2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> 2
[3,1,2,4] => [3,1]
=> [1]
=> 1
[3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> [1,1]
=> 2
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> 2
[4,1,2,3] => [3,1]
=> [1]
=> 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> [1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> [1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,4,5] => [5]
=> []
=> 0
[1,2,3,5,4] => [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [4,1]
=> [1]
=> 1
[1,2,4,5,3] => [4,1]
=> [1]
=> 1
[1,2,5,3,4] => [4,1]
=> [1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [4,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [4,1]
=> [1]
=> 1
[1,4,2,5,3] => [3,2]
=> [2]
=> 2
[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] => [3,2]
=> [2]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000377
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer 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] => [2]
=> [1,1]
=> [2]
=> 0
[2,1] => [1,1]
=> [2]
=> [1,1]
=> 1
[1,2,3] => [3]
=> [1,1,1]
=> [2,1]
=> 0
[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] => [1,1,1]
=> [3]
=> [1,1,1]
=> 2
[1,2,3,4] => [4]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,2,4,3] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,3,2,4] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[2,1,3,4] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [4]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,3,4,1] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,4,1,3] => [2,2]
=> [2,2]
=> [4]
=> 2
[2,4,3,1] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,1,4,2] => [2,2]
=> [2,2]
=> [4]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> [2,2]
=> [4]
=> 2
[3,4,2,1] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,1,2,3] => [3,1]
=> [2,1,1]
=> [2,2]
=> 1
[4,1,3,2] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 3
[1,2,3,4,5] => [5]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [3,2]
=> [2,2,1]
=> [2,2,1]
=> 2
[1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,5,2,4] => [3,2]
=> [2,2,1]
=> [2,2,1]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [4,1]
=> 2
[1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[1,4,2,5,3] => [3,2]
=> [2,2,1]
=> [2,2,1]
=> 2
[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] => [3,2]
=> [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 $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001034
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck 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] => [2]
=> []
=> []
=> 0
[2,1] => [1,1]
=> [1]
=> [1,0]
=> 1
[1,2,3] => [3]
=> []
=> []
=> 0
[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] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,2,3,4] => [4]
=> []
=> []
=> 0
[1,2,4,3] => [3,1]
=> [1]
=> [1,0]
=> 1
[1,3,2,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> 1
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1,3,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[2,3,4,1] => [3,1]
=> [1]
=> [1,0]
=> 1
[2,4,1,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> 1
[3,1,4,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,1,2,3] => [3,1]
=> [1]
=> [1,0]
=> 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,2,3,4,5] => [5]
=> []
=> []
=> 0
[1,2,3,5,4] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,2,4,3,5] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,2,4,5,3] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,2,5,3,4] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,3,2,4,5] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,3,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,3,4,2,5] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,3,5,2,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,4,2,3,5] => [4,1]
=> [1]
=> [1,0]
=> 1
[1,4,2,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[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] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000293
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 100%●distinct values known / distinct values provided: 91%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary 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] => [2]
=> []
=> => ? = 0
[2,1] => [1,1]
=> [1]
=> 10 => 1
[1,2,3] => [3]
=> []
=> => ? = 0
[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] => [1,1,1]
=> [1,1]
=> 110 => 2
[1,2,3,4] => [4]
=> []
=> => ? = 0
[1,2,4,3] => [3,1]
=> [1]
=> 10 => 1
[1,3,2,4] => [3,1]
=> [1]
=> 10 => 1
[1,3,4,2] => [3,1]
=> [1]
=> 10 => 1
[1,4,2,3] => [3,1]
=> [1]
=> 10 => 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 110 => 2
[2,1,3,4] => [3,1]
=> [1]
=> 10 => 1
[2,1,4,3] => [2,2]
=> [2]
=> 100 => 2
[2,3,1,4] => [3,1]
=> [1]
=> 10 => 1
[2,3,4,1] => [3,1]
=> [1]
=> 10 => 1
[2,4,1,3] => [2,2]
=> [2]
=> 100 => 2
[2,4,3,1] => [2,1,1]
=> [1,1]
=> 110 => 2
[3,1,2,4] => [3,1]
=> [1]
=> 10 => 1
[3,1,4,2] => [2,2]
=> [2]
=> 100 => 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 110 => 2
[3,2,4,1] => [2,1,1]
=> [1,1]
=> 110 => 2
[3,4,1,2] => [2,2]
=> [2]
=> 100 => 2
[3,4,2,1] => [2,1,1]
=> [1,1]
=> 110 => 2
[4,1,2,3] => [3,1]
=> [1]
=> 10 => 1
[4,1,3,2] => [2,1,1]
=> [1,1]
=> 110 => 2
[4,2,1,3] => [2,1,1]
=> [1,1]
=> 110 => 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 110 => 2
[4,3,1,2] => [2,1,1]
=> [1,1]
=> 110 => 2
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1110 => 3
[1,2,3,4,5] => [5]
=> []
=> => ? = 0
[1,2,3,5,4] => [4,1]
=> [1]
=> 10 => 1
[1,2,4,3,5] => [4,1]
=> [1]
=> 10 => 1
[1,2,4,5,3] => [4,1]
=> [1]
=> 10 => 1
[1,2,5,3,4] => [4,1]
=> [1]
=> 10 => 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,3,2,4,5] => [4,1]
=> [1]
=> 10 => 1
[1,3,2,5,4] => [3,2]
=> [2]
=> 100 => 2
[1,3,4,2,5] => [4,1]
=> [1]
=> 10 => 1
[1,3,4,5,2] => [4,1]
=> [1]
=> 10 => 1
[1,3,5,2,4] => [3,2]
=> [2]
=> 100 => 2
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,4,2,3,5] => [4,1]
=> [1]
=> 10 => 1
[1,4,2,5,3] => [3,2]
=> [2]
=> 100 => 2
[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] => [3,2]
=> [2]
=> 100 => 2
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,5,2,3,4] => [4,1]
=> [1]
=> 10 => 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 110 => 2
[1,2,3,4,5,6] => [6]
=> []
=> => ? = 0
[1,2,3,4,5,6,7] => [7]
=> []
=> => ? = 0
[1,2,3,4,5,6,7,8] => [8]
=> []
=> => ? = 0
[1,2,3,4,5,6,7,8,9] => [9]
=> []
=> => ? = 0
[1,2,3,4,5,6,7,8,9,10] => [10]
=> []
=> => ? = 0
[1,2,3,4,5,6,7,8,9,10,11] => [11]
=> []
=> => ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000738
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: 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] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,1] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,3] => [3]
=> [[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] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,1,3,4] => [3,1]
=> [[1,3,4],[2]]
=> 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]]
=> 2 = 1 + 1
[2,3,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,2,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,1,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[4,1,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,4,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,5,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,3,4,2,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,4,2,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 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] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,4,3,6,5,10,9,8,7,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,4,3,6,5,12,9,8,11,10,7] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,4,3,6,5,12,11,10,9,8,7] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,4,3,8,7,6,5,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,4,3,10,7,6,9,8,5,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,4,3,12,7,6,9,8,11,10,5] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,4,3,12,7,6,11,10,9,8,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,4,3,10,9,8,7,6,5,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,4,3,12,9,8,7,6,11,10,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,4,3,12,11,8,7,10,9,6,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,4,3,12,11,10,9,8,7,6,5] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,1,6,5,4,3,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,8,5,4,7,6,3,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,10,5,4,7,6,9,8,3,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,12,5,4,7,6,9,8,11,10,3] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[2,1,12,5,4,7,6,11,10,9,8,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,10,5,4,9,8,7,6,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,12,5,4,9,8,7,6,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,12,5,4,11,8,7,10,9,6,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,12,5,4,11,10,9,8,7,6,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,1,8,7,6,5,4,3,10,9,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,10,7,6,5,4,9,8,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,12,7,6,5,4,9,8,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,10,9,6,5,8,7,4,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,12,9,6,5,8,7,4,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,12,11,6,5,8,7,10,9,4,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[2,1,12,11,6,5,10,9,8,7,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,1,10,9,8,7,6,5,4,3,12,11] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,1,12,9,8,7,6,5,4,11,10,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,1,12,11,8,7,6,5,10,9,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,1,12,11,10,7,6,9,8,5,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[2,1,12,11,10,9,8,7,6,5,4,3] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 10 + 1
[4,3,2,1,6,5,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[4,3,2,1,8,7,6,5,12,11,10,9] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 9 + 1
[4,3,2,1,12,11,10,9,8,7,6,5] => [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ? = 10 + 1
[6,3,2,5,4,1,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[8,3,2,5,4,7,6,1,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[10,3,2,5,4,7,6,9,8,1,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[12,3,2,5,4,7,6,9,8,11,10,1] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 7 + 1
[12,3,2,5,4,7,6,11,10,9,8,1] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[10,3,2,5,4,9,8,7,6,1,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[12,3,2,5,4,9,8,7,6,11,10,1] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[12,3,2,5,4,11,8,7,10,9,6,1] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[12,3,2,5,4,11,10,9,8,7,6,1] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 9 + 1
[8,3,2,7,6,5,4,1,10,9,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[10,3,2,7,6,5,4,9,8,1,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[12,3,2,7,6,5,4,9,8,11,10,1] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[10,3,2,9,6,5,8,7,4,1,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
[12,3,2,9,6,5,8,7,4,11,10,1] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 8 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000507
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[2,1] => [1,1]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[1,2,3] => [3]
=> [1,1,1]
=> [[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] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,2,3,4] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,3,2,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[2,1,3,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 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]]
=> 2 = 1 + 1
[2,3,4,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[2,4,1,3] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[3,1,4,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,2,4,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,1,2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,1,3,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,2,1,3] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,3,4,5] => [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,5,4] => [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 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] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,4,2,5,3] => [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[1,4,3,2,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 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] => [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,4,3,6,5,10,9,8,7,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,4,3,6,5,12,9,8,11,10,7] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,4,3,6,5,12,11,10,9,8,7] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,4,3,8,7,6,5,10,9,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,4,3,10,7,6,9,8,5,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,4,3,12,7,6,9,8,11,10,5] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,4,3,12,7,6,11,10,9,8,5] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,4,3,10,9,8,7,6,5,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,4,3,12,9,8,7,6,11,10,5] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,4,3,12,11,8,7,10,9,6,5] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,4,3,12,11,10,9,8,7,6,5] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,1,6,5,4,3,8,7,10,9,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,8,5,4,7,6,3,10,9,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,10,5,4,7,6,9,8,3,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,12,5,4,7,6,9,8,11,10,3] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[2,1,12,5,4,7,6,11,10,9,8,3] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,10,5,4,9,8,7,6,3,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,12,5,4,9,8,7,6,11,10,3] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,12,5,4,11,8,7,10,9,6,3] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,12,5,4,11,10,9,8,7,6,3] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,1,8,7,6,5,4,3,10,9,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,10,7,6,5,4,9,8,3,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,12,7,6,5,4,9,8,11,10,3] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,10,9,6,5,8,7,4,3,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,12,9,6,5,8,7,4,11,10,3] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,12,11,6,5,8,7,10,9,4,3] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[2,1,12,11,6,5,10,9,8,7,4,3] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,1,10,9,8,7,6,5,4,3,12,11] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,1,12,9,8,7,6,5,4,11,10,3] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,1,12,11,8,7,6,5,10,9,4,3] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,1,12,11,10,7,6,9,8,5,4,3] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[2,1,12,11,10,9,8,7,6,5,4,3] => [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? = 10 + 1
[4,3,2,1,6,5,8,7,10,9,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[4,3,2,1,8,7,6,5,12,11,10,9] => [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 9 + 1
[4,3,2,1,12,11,10,9,8,7,6,5] => [2,2,2,2,1,1,1,1]
=> [8,4]
=> [[1,2,3,4,9,10,11,12],[5,6,7,8]]
=> ? = 10 + 1
[6,3,2,5,4,1,8,7,10,9,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[8,3,2,5,4,7,6,1,10,9,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[10,3,2,5,4,7,6,9,8,1,12,11] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[12,3,2,5,4,7,6,9,8,11,10,1] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 7 + 1
[12,3,2,5,4,7,6,11,10,9,8,1] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[10,3,2,5,4,9,8,7,6,1,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[12,3,2,5,4,9,8,7,6,11,10,1] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[12,3,2,5,4,11,8,7,10,9,6,1] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[12,3,2,5,4,11,10,9,8,7,6,1] => [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 9 + 1
[8,3,2,7,6,5,4,1,10,9,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[10,3,2,7,6,5,4,9,8,1,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[12,3,2,7,6,5,4,9,8,11,10,1] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[10,3,2,9,6,5,8,7,4,1,12,11] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 1
[12,3,2,9,6,5,8,7,4,11,10,1] => [4,4,1,1,1,1]
=> [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? = 8 + 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
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 0 + 1
[2,1] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 1 + 1
[1,2,3] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 0 + 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] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,1,3,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,3,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,2,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,1,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,1,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,2,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,4,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,4,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,5,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 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] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3 = 2 + 1
[1,3,4,2,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3 = 2 + 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] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,4,2,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3 = 2 + 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] => [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3 = 2 + 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 1
[2,1,4,3,6,5,10,9,8,7,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 1
[2,1,4,3,6,5,12,9,8,11,10,7] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 1
[2,1,4,3,6,5,12,11,10,9,8,7] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,4,3,8,7,6,5,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 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] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 1
[2,1,4,3,12,7,6,9,8,11,10,5] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 1
[2,1,4,3,12,7,6,11,10,9,8,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,4,3,10,9,8,7,6,5,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,4,3,12,9,8,7,6,11,10,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,4,3,12,11,8,7,10,9,6,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,4,3,12,11,10,9,8,7,6,5] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 9 + 1
[2,1,6,5,4,3,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 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] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 9 + 1
[2,1,8,5,4,7,6,3,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 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] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 1
[2,1,12,5,4,7,6,9,8,11,10,3] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 1
[2,1,12,5,4,7,6,11,10,9,8,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,10,5,4,9,8,7,6,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,5,4,9,8,7,6,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,5,4,11,8,7,10,9,6,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,5,4,11,10,9,8,7,6,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 9 + 1
[2,1,8,7,6,5,4,3,10,9,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,8,7,6,5,4,3,12,11,10,9] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 9 + 1
[2,1,10,7,6,5,4,9,8,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,7,6,5,4,9,8,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,7,6,5,4,11,10,9,8,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 9 + 1
[2,1,10,9,6,5,8,7,4,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,9,6,5,8,7,4,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,11,6,5,8,7,10,9,4,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ?
=> ? = 8 + 1
[2,1,12,11,6,5,10,9,8,7,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 9 + 1
[2,1,10,9,8,7,6,5,4,3,12,11] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 9 + 1
[2,1,12,9,8,7,6,5,4,11,10,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 9 + 1
[2,1,12,11,8,7,6,5,10,9,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 9 + 1
[2,1,12,11,10,7,6,9,8,5,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ?
=> ? = 9 + 1
[2,1,12,11,10,9,8,7,6,5,4,3] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 10 + 1
[4,3,2,1,6,5,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 7 + 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] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 9 + 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] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 9 + 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
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 92%●distinct values known / distinct values provided: 91%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 92%●distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[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] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,3,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,1,3,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,3,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,4,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,2,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,1,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4,1,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,2,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,3,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,4,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,4,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,5,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,3,2,4,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,2,5,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,3,4,2,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,3,5,2,4] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
[1,4,2,3,5] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,4,2,5,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[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] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2
[2,1,4,3,6,5,8,7,12,11,10,9] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[2,1,4,3,6,5,10,9,8,7,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[2,1,4,3,6,5,12,9,8,11,10,7] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[2,1,4,3,6,5,12,11,10,9,8,7] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,4,3,8,7,6,5,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[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] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[2,1,4,3,12,7,6,9,8,11,10,5] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[2,1,4,3,12,7,6,11,10,9,8,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,4,3,10,9,8,7,6,5,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,4,3,12,9,8,7,6,11,10,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,4,3,12,11,8,7,10,9,6,5] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,4,3,12,11,10,9,8,7,6,5] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? => ? = 9
[2,1,6,5,4,3,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[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] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [10,7,5,11,3,8,2,6,12,1,4,9] => ? = 9
[2,1,8,5,4,7,6,3,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[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] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[2,1,12,5,4,7,6,9,8,11,10,3] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[2,1,12,5,4,7,6,11,10,9,8,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,10,5,4,9,8,7,6,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,5,4,9,8,7,6,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,5,4,11,8,7,10,9,6,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,5,4,11,10,9,8,7,6,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? => ? = 9
[2,1,8,7,6,5,4,3,10,9,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,8,7,6,5,4,3,12,11,10,9] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [10,7,5,11,3,8,2,6,12,1,4,9] => ? = 9
[2,1,10,7,6,5,4,9,8,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,7,6,5,4,9,8,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,7,6,5,4,11,10,9,8,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [10,7,5,11,3,8,2,6,12,1,4,9] => ? = 9
[2,1,10,9,6,5,8,7,4,3,12,11] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,9,6,5,8,7,4,11,10,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,11,6,5,8,7,10,9,4,3] => [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? = 8
[2,1,12,11,6,5,10,9,8,7,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? => ? = 9
[2,1,10,9,8,7,6,5,4,3,12,11] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? => ? = 9
[2,1,12,9,8,7,6,5,4,11,10,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? => ? = 9
[2,1,12,11,8,7,6,5,10,9,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? => ? = 9
[2,1,12,11,10,7,6,9,8,5,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? => ? = 9
[2,1,12,11,10,9,8,7,6,5,4,3] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? => ? = 10
[4,3,2,1,6,5,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> [8,3,2,9,10,11,12,1,4,5,6,7] => ? = 7
[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] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [10,7,5,11,3,8,2,6,12,1,4,9] => ? = 9
[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] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 9
[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
[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]]
=> [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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 92%●distinct values known / distinct values provided: 91%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 92%●distinct values known / distinct values provided: 91%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,2,3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 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] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,2,3,4] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,3,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[2,1,3,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 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] => 1
[2,3,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[2,4,1,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,4,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[3,2,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[4,1,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,1,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[4,2,1,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[4,3,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,4,3,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,4,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,5,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,3,2,4,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[1,3,4,2,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,2,3,5] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,4,2,5,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[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] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,1,4,3,6,5,8,7,12,11,10,9] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[2,1,4,3,6,5,10,9,8,7,12,11] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[2,1,4,3,6,5,12,9,8,11,10,7] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[2,1,4,3,6,5,12,11,10,9,8,7] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,4,3,8,7,6,5,10,9,12,11] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[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] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[2,1,4,3,12,7,6,9,8,11,10,5] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[2,1,4,3,12,7,6,11,10,9,8,5] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,4,3,10,9,8,7,6,5,12,11] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,4,3,12,9,8,7,6,11,10,5] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,4,3,12,11,8,7,10,9,6,5] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,4,3,12,11,10,9,8,7,6,5] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 9
[2,1,6,5,4,3,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[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] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 9
[2,1,8,5,4,7,6,3,10,9,12,11] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[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] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[2,1,12,5,4,7,6,9,8,11,10,3] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[2,1,12,5,4,7,6,11,10,9,8,3] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,10,5,4,9,8,7,6,3,12,11] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,5,4,9,8,7,6,11,10,3] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,5,4,11,8,7,10,9,6,3] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,5,4,11,10,9,8,7,6,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 9
[2,1,8,7,6,5,4,3,10,9,12,11] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,8,7,6,5,4,3,12,11,10,9] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 9
[2,1,10,7,6,5,4,9,8,3,12,11] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,7,6,5,4,9,8,11,10,3] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,7,6,5,4,11,10,9,8,3] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 9
[2,1,10,9,6,5,8,7,4,3,12,11] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,9,6,5,8,7,4,11,10,3] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,11,6,5,8,7,10,9,4,3] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? => ? = 8
[2,1,12,11,6,5,10,9,8,7,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 9
[2,1,10,9,8,7,6,5,4,3,12,11] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 9
[2,1,12,9,8,7,6,5,4,11,10,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 9
[2,1,12,11,8,7,6,5,10,9,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 9
[2,1,12,11,10,7,6,9,8,5,4,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 9
[2,1,12,11,10,9,8,7,6,5,4,3] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? = 10
[4,3,2,1,6,5,8,7,10,9,12,11] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? => ? = 7
[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] => [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 9
[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] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 9
[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
[4,3,2,1,12,7,6,9,8,11,10,5] => [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 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St001726The number of visible inversions of a permutation. 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. St000839The largest opener of a set partition. St000288The number of ones in a binary word. St000157The number of descents of a standard tableau. St000024The number of double up and double down steps of a Dyck path. St000211The rank of the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000362The size of a minimal vertex cover of a graph. St000245The number of ascents of a permutation. St000703The number of deficiencies of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000470The number of runs in a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000740The last entry of a permutation. St000209Maximum difference of elements in cycles. St000021The number of descents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000325The width of the tree associated to a permutation. St000051The size of the left subtree of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000956The maximal displacement 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. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001427The number of descents of a signed 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000454The largest eigenvalue of a graph if it is integral. St001626The number of maximal proper sublattices of a lattice.
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