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Your data matches 56 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)
Mp00204: Permutations —LLPS⟶ 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] => [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.
Matching statistic: St000377
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ 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] => [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 —LLPS⟶ 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] => [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].
Matching statistic: St000228
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
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
[1,2,3,4,5,6,7,8,9,10,11,12] => [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 100%●distinct values known / distinct values provided: 83%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 100%●distinct values known / distinct values provided: 83%
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
[1,2,3,4,5,6,7,8,9,10,11,12] => [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[11,10,9,8,7,6,5,4,3,2,1] => [11]
=> []
=> => ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000738
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 92%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 92%
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]].
Matching statistic: St000507
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard 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 —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●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: 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 —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 92%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 92%
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: St000054
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 92%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 92%
Values
[1] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2 = 1 + 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 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]]
=> [4,3,2,1] => 4 = 3 + 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[1,3,4,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[1,4,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[1,4,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 3 = 2 + 1
[2,3,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[2,3,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[2,4,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[3,1,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 3 = 2 + 1
[3,2,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[3,4,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[3,4,2,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[4,1,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3 = 2 + 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[4,2,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2 = 1 + 1
[4,3,1,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 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]]
=> [5,4,3,2,1] => 5 = 4 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,2,4,5,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,2,5,3,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3 = 2 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 4 = 3 + 1
[1,3,4,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,3,4,5,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,3,5,2,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3 = 2 + 1
[1,4,2,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,4,2,5,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 4 = 3 + 1
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3 = 2 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3 = 2 + 1
[1,4,5,2,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4 = 3 + 1
[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 + 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 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]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 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]]
=> ? => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 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]]
=> [9,10,5,6,3,4,11,12,1,2,7,8] => ? = 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
[4,3,2,1,10,7,6,9,8,5,12,11] => [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 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000662The staircase size of the code of a 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. St001971The number of negative eigenvalues of the adjacency matrix of the 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.
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