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Your data matches 103 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 2
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 3
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 3
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 4
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 2
[3,3]
=> 3
[3,2,1]
=> 3
[3,1,1,1]
=> 3
[2,2,2]
=> 4
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 5
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 2
[4,3]
=> 3
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 4
[3,2,2]
=> 4
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 5
[2,2,1,1,1]
=> 5
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 6
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 2
[5,3]
=> 3
[5,2,1]
=> 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: St001034
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> 0
[2]
=> []
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[4]
=> []
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[5]
=> []
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[2,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[6]
=> []
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[3,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[3,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
[7]
=> []
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[4,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[4,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[3,2,2]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[8]
=> []
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[5,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[5,2,1]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 11
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 11
[4,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 11
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 11
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 11
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 11
[4,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 11
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 11
[3,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 12
[2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 13
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 13
[5,5,4,1,1]
=> [5,4,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 11
[5,5,3,3]
=> [5,3,3]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 11
[5,5,3,2,1]
=> [5,3,2,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 11
[5,5,2,2,1,1]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 11
[5,5,2,1,1,1,1]
=> [5,2,1,1,1,1]
=> [6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 11
[5,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 11
[5,4,4,1,1,1]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 11
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 11
[5,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 11
[5,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 11
[5,3,3,2,2,1]
=> [3,3,2,2,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 11
[4,4,4,2,2]
=> [4,4,2,2]
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 12
[4,4,4,2,1,1]
=> [4,4,2,1,1]
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 12
[4,4,4,1,1,1,1]
=> [4,4,1,1,1,1]
=> [8,3,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 12
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 12
[4,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> [6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 12
[4,4,3,1,1,1,1,1]
=> [4,3,1,1,1,1,1]
=> [7,3,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 12
[4,4,2,2,2,1,1]
=> [4,2,2,2,1,1]
=> [6,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 12
[4,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
[4,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
[4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 12
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 13
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,3,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 13
[3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 13
[3,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 13
[3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 13
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [6,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 14
[2,2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,2,1,1]
=> [6,5,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 14
[6,6,5]
=> [6,5]
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
[6,6,3,2]
=> [6,3,2]
=> [7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 11
[6,6,3,1,1]
=> [6,3,1,1]
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 11
[6,6,2,2,1]
=> [6,2,2,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 11
[6,6,2,1,1,1]
=> [6,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 11
[6,6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 11
[6,5,4,1,1]
=> [5,4,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 11
[6,5,3,3]
=> [5,3,3]
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 11
[6,5,3,2,1]
=> [5,3,2,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 11
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 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 94%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 94%
Values
[1]
=> []
=> []
=> => ? = 0
[2]
=> []
=> []
=> => ? = 0
[1,1]
=> [1]
=> [1]
=> 10 => 1
[3]
=> []
=> []
=> => ? = 0
[2,1]
=> [1]
=> [1]
=> 10 => 1
[1,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[4]
=> []
=> []
=> => ? = 0
[3,1]
=> [1]
=> [1]
=> 10 => 1
[2,2]
=> [2]
=> [1,1]
=> 110 => 2
[2,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[5]
=> []
=> []
=> => ? = 0
[4,1]
=> [1]
=> [1]
=> 10 => 1
[3,2]
=> [2]
=> [1,1]
=> 110 => 2
[3,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[2,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[6]
=> []
=> []
=> => ? = 0
[5,1]
=> [1]
=> [1]
=> 10 => 1
[4,2]
=> [2]
=> [1,1]
=> 110 => 2
[4,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[3,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[3,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[2,2,2]
=> [2,2]
=> [4]
=> 10000 => 4
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 10100 => 5
[7]
=> []
=> []
=> => ? = 0
[6,1]
=> [1]
=> [1]
=> 10 => 1
[5,2]
=> [2]
=> [1,1]
=> 110 => 2
[5,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[4,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[4,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 4
[3,2,2]
=> [2,2]
=> [4]
=> 10000 => 4
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 11010 => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 100110 => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 10100 => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 101010 => 6
[8]
=> []
=> []
=> => ? = 0
[7,1]
=> [1]
=> [1]
=> 10 => 1
[6,2]
=> [2]
=> [1,1]
=> 110 => 2
[6,1,1]
=> [1,1]
=> [2]
=> 100 => 2
[5,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[5,2,1]
=> [2,1]
=> [3]
=> 1000 => 3
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 3
[4,4]
=> [4]
=> [1,1,1,1]
=> 11110 => 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 4
[4,2,2]
=> [2,2]
=> [4]
=> 10000 => 4
[4,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 4
[4,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 10010 => 4
[3,3,2]
=> [3,2]
=> [5]
=> 100000 => 5
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> 100010 => 5
[9]
=> []
=> []
=> => ? = 0
[10]
=> []
=> []
=> => ? = 0
[11]
=> []
=> []
=> => ? = 0
[12]
=> []
=> []
=> => ? = 0
[13]
=> []
=> []
=> => ? = 0
[14]
=> []
=> []
=> => ? = 0
[3,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 11
[15]
=> []
=> []
=> => ? = 0
[4,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> 1000001000 => ? = 11
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> 1000001100 => ? = 11
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? = 11
[4,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 11
[3,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> 1000000111110 => ? = 12
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> 100001011110 => ? = 12
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> 10000100110 => ? = 12
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 12
[2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> 10000111010 => ? = 13
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> 10010001110 => ? = 13
[16]
=> []
=> []
=> => ? = 0
[5,5,5,1]
=> [5,5,1]
=> [10,1]
=> 100000000010 => ? = 11
[5,5,4,2]
=> [5,4,2]
=> [9,1,1]
=> 100000000110 => ? = 11
[5,5,3,3]
=> [5,3,3]
=> [8,1,1,1]
=> 100000001110 => ? = 11
[5,5,2,2,1,1]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> 1000011010 => ? = 11
[5,5,2,1,1,1,1]
=> [5,2,1,1,1,1]
=> [6,3,1,1]
=> 1000100110 => ? = 11
[5,4,4,2,1]
=> [4,4,2,1]
=> [8,3]
=> 1000001000 => ? = 11
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> 1000001100 => ? = 11
[5,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? = 11
[5,3,3,3,2]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 11
[4,4,4,2,2]
=> [4,4,2,2]
=> [8,4]
=> 1000010000 => ? = 12
[4,4,4,2,1,1]
=> [4,4,2,1,1]
=> [8,2,2]
=> 10000001100 => ? = 12
[4,4,4,1,1,1,1]
=> [4,4,1,1,1,1]
=> [8,3,1]
=> 10000010010 => ? = 12
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [6,2,2,1,1]
=> 10000110110 => ? = 12
[4,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> [6,3,1,1,1]
=> 10001001110 => ? = 12
[4,4,3,1,1,1,1,1]
=> [4,3,1,1,1,1,1]
=> [7,3,2]
=> 1000010100 => ? = 12
[4,4,2,2,2,1,1]
=> [4,2,2,2,1,1]
=> [6,2,2,2]
=> 1000011100 => ? = 12
[4,3,3,3,3]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> 1000000111110 => ? = 12
[4,3,3,3,2,1]
=> [3,3,3,2,1]
=> [6,2,1,1,1,1]
=> 100001011110 => ? = 12
[4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> 10000100110 => ? = 12
[4,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 12
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> 1000001011110 => ? = 13
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,3,1,1,1,1]
=> 100010011110 => ? = 13
[3,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [7,5,1]
=> 1001000010 => ? = 13
Description
The number of inversions of a binary word.
Matching statistic: St000290
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 88%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 88%
Values
[1]
=> []
=> => ? => ? = 0
[2]
=> []
=> => ? => ? = 0
[1,1]
=> [1]
=> 10 => 10 => 1
[3]
=> []
=> => ? => ? = 0
[2,1]
=> [1]
=> 10 => 10 => 1
[1,1,1]
=> [1,1]
=> 110 => 110 => 2
[4]
=> []
=> => ? => ? = 0
[3,1]
=> [1]
=> 10 => 10 => 1
[2,2]
=> [2]
=> 100 => 010 => 2
[2,1,1]
=> [1,1]
=> 110 => 110 => 2
[1,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[5]
=> []
=> => ? => ? = 0
[4,1]
=> [1]
=> 10 => 10 => 1
[3,2]
=> [2]
=> 100 => 010 => 2
[3,1,1]
=> [1,1]
=> 110 => 110 => 2
[2,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[2,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[6]
=> []
=> => ? => ? = 0
[5,1]
=> [1]
=> 10 => 10 => 1
[4,2]
=> [2]
=> 100 => 010 => 2
[4,1,1]
=> [1,1]
=> 110 => 110 => 2
[3,3]
=> [3]
=> 1000 => 0010 => 3
[3,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[3,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[2,2,2]
=> [2,2]
=> 1100 => 1010 => 4
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 5
[7]
=> []
=> => ? => ? = 0
[6,1]
=> [1]
=> 10 => 10 => 1
[5,2]
=> [2]
=> 100 => 010 => 2
[5,1,1]
=> [1,1]
=> 110 => 110 => 2
[4,3]
=> [3]
=> 1000 => 0010 => 3
[4,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[4,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[3,3,1]
=> [3,1]
=> 10010 => 00110 => 4
[3,2,2]
=> [2,2]
=> 1100 => 1010 => 4
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[2,2,2,1]
=> [2,2,1]
=> 11010 => 10110 => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 011110 => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => 6
[8]
=> []
=> => ? => ? = 0
[7,1]
=> [1]
=> 10 => 10 => 1
[6,2]
=> [2]
=> 100 => 010 => 2
[6,1,1]
=> [1,1]
=> 110 => 110 => 2
[5,3]
=> [3]
=> 1000 => 0010 => 3
[5,2,1]
=> [2,1]
=> 1010 => 0110 => 3
[5,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 3
[4,4]
=> [4]
=> 10000 => 00010 => 4
[4,3,1]
=> [3,1]
=> 10010 => 00110 => 4
[4,2,2]
=> [2,2]
=> 1100 => 1010 => 4
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 4
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 4
[3,3,2]
=> [3,2]
=> 10100 => 10010 => 5
[3,3,1,1]
=> [3,1,1]
=> 100110 => 001110 => 5
[9]
=> []
=> => ? => ? = 0
[10]
=> []
=> => ? => ? = 0
[11]
=> []
=> => ? => ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? = 9
[12]
=> []
=> => ? => ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? = 9
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? = 9
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111110 => ? = 10
[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
[13]
=> []
=> => ? => ? = 0
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? = 9
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? = 9
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => 00111111110 => ? = 10
[3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111110 => ? = 10
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => ? => ? = 11
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => ? => ? = 11
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 11
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 11
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? = 12
[14]
=> []
=> => ? => ? = 0
[5,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? = 9
[5,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? = 9
[4,4,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1]
=> 10001111110 => 00011111110 => ? = 10
[4,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => 00111111110 => ? = 10
[4,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01111111110 => ? = 10
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? = 11
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? = 11
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => ? => ? = 11
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? = 11
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => ? => ? = 11
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => ? => ? = 11
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 11
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 11
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? => ? = 12
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? => ? = 12
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? => ? = 12
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? => ? = 12
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111110 => ? => ? = 13
[15]
=> []
=> => ? => ? = 0
[6,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0011111110 => ? = 9
[6,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0111111110 => ? = 9
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000246
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 88%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 88%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[5]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 5
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 6
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? = 11
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ? = 11
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,9,8,4,7,6,5,3,2] => ? = 11
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,5,7,6,4,3,2] => ? = 11
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,6,7,5,4,3,2] => ? = 11
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 12
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? = 11
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => ? = 11
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ? = 11
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,8,4,7,5,6,3,2] => ? = 11
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,9,8,5,6,7,4,3,2] => ? = 11
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,10,9,8,6,5,7,4,3,2] => ? = 11
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? = 11
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ? = 11
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,9,8,4,7,6,5,3,2] => ? = 11
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,5,7,6,4,3,2] => ? = 11
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,6,7,5,4,3,2] => ? = 11
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,8,7,6,5,4,2] => ? = 12
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,9,4,8,7,6,5,3,2] => ? = 12
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,5,8,7,6,4,3,2] => ? = 12
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,6,8,7,5,4,3,2] => ? = 12
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,12,11,10,9,7,8,6,5,4,3,2] => ? = 12
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 13
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,7,4,3,5,6,2] => ? = 11
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,7,3,5,4,6,2] => ? = 11
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,8,6,4,5,7,3,2] => ? = 11
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,9,8,6,5,4,7,3,2] => ? = 11
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? = 11
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => ? = 11
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ? = 11
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,8,4,7,5,6,3,2] => ? = 11
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,9,8,5,6,7,4,3,2] => ? = 11
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,10,9,8,6,5,7,4,3,2] => ? = 11
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? = 11
[4,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ? = 11
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,9,8,4,7,6,5,3,2] => ? = 11
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,5,7,6,4,3,2] => ? = 11
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,6,7,5,4,3,2] => ? = 11
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 12
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,3,7,6,5,2] => ? = 12
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,7,4,6,5,3] => ? = 12
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => ? = 12
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,8,4,5,7,6,3,2] => ? = 12
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,9,8,5,4,7,6,3,2] => ? = 12
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => ? = 12
Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000018
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 88%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 88%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[5]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 11
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? = 11
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? = 11
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? = 11
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? = 11
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 12
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? = 11
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => ? = 11
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? = 11
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? = 11
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? = 11
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => ? = 11
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => ? = 11
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 11
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? = 11
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? = 11
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? = 11
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? = 11
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,2,8,1] => ? = 12
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,9,1] => ? = 12
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,10,1] => ? = 12
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,11,1] => ? = 12
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,12,1] => ? = 12
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 13
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => ? = 11
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => ? = 11
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => ? = 11
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,1] => ? = 11
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? = 11
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => ? = 11
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? = 11
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? = 11
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? = 11
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => ? = 11
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => ? = 11
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 11
[4,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? = 11
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? = 11
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? = 11
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? = 11
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 12
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => ? = 12
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,1,2] => ? = 12
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => ? = 12
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => ? = 12
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000228
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 72%●distinct values known / distinct values provided: 69%
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 72%●distinct values known / distinct values provided: 69%
Values
[1]
=> []
=> 0
[2]
=> []
=> 0
[1,1]
=> [1]
=> 1
[3]
=> []
=> 0
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 2
[4]
=> []
=> 0
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 2
[1,1,1,1]
=> [1,1,1]
=> 3
[5]
=> []
=> 0
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 2
[2,2,1]
=> [2,1]
=> 3
[2,1,1,1]
=> [1,1,1]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> 4
[6]
=> []
=> 0
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 2
[3,3]
=> [3]
=> 3
[3,2,1]
=> [2,1]
=> 3
[3,1,1,1]
=> [1,1,1]
=> 3
[2,2,2]
=> [2,2]
=> 4
[2,2,1,1]
=> [2,1,1]
=> 4
[2,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[7]
=> []
=> 0
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 2
[4,3]
=> [3]
=> 3
[4,2,1]
=> [2,1]
=> 3
[4,1,1,1]
=> [1,1,1]
=> 3
[3,3,1]
=> [3,1]
=> 4
[3,2,2]
=> [2,2]
=> 4
[3,2,1,1]
=> [2,1,1]
=> 4
[3,1,1,1,1]
=> [1,1,1,1]
=> 4
[2,2,2,1]
=> [2,2,1]
=> 5
[2,2,1,1,1]
=> [2,1,1,1]
=> 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 6
[8]
=> []
=> 0
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 2
[5,3]
=> [3]
=> 3
[5,2,1]
=> [2,1]
=> 3
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> ? = 11
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> ? = 11
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> ? = 11
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 11
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[3,3,3,3,2]
=> [3,3,3,2]
=> ? = 11
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> ? = 11
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? = 11
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> ? = 11
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> ? = 11
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> ? = 11
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> ? = 11
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> ? = 11
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> ? = 11
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 11
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> ? = 11
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> ? = 11
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> ? = 11
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 11
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 11
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> ? = 12
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> ? = 12
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> ? = 12
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 12
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> ? = 12
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
[4,4,4,3]
=> [4,4,3]
=> ? = 11
[4,4,4,2,1]
=> [4,4,2,1]
=> ? = 11
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> ? = 11
[4,4,3,3,1]
=> [4,3,3,1]
=> ? = 11
[4,4,3,2,2]
=> [4,3,2,2]
=> ? = 11
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> ? = 11
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> ? = 11
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> ? = 11
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> ? = 11
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 11
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> ? = 11
[4,3,3,3,2]
=> [3,3,3,2]
=> ? = 11
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> ? = 11
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> ? = 11
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> ? = 11
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> ? = 11
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> ? = 11
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> ? = 11
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001759
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 88%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 88%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> []
=> []
=> [] => ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4]
=> []
=> []
=> [] => ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[5]
=> []
=> []
=> [] => ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[6]
=> []
=> []
=> [] => ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[7]
=> []
=> []
=> [] => ? = 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[8]
=> []
=> []
=> [] => ? = 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 7
[9]
=> []
=> []
=> [] => ? = 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 7
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 8
[10]
=> []
=> []
=> [] => ? = 0
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 7
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 8
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 8
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 9
[11]
=> []
=> []
=> [] => ? = 0
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 7
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 8
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 8
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? = 9
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? = 9
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? = 10
[12]
=> []
=> []
=> [] => ? = 0
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 7
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 8
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 8
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? = 9
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? = 9
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? = 9
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 9
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => ? = 10
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => ? = 10
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => ? = 10
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 11
[13]
=> []
=> []
=> [] => ? = 0
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
[6,6,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 7
[6,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 7
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 8
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 8
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? = 9
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? = 9
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? = 9
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 9
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => ? = 10
Description
The Rajchgot index of a permutation.
The '''Rajchgot index''' of a permutation $\sigma$ is the degree of the ''Grothendieck polynomial'' of $\sigma$. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to $\sigma$ in the right ''weak Bruhat order''.
Matching statistic: St000395
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 88%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 88%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[9]
=> []
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[11]
=> []
=> []
=> []
=> ? = 0
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[12]
=> []
=> []
=> []
=> ? = 0
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 8
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 10
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
[13]
=> []
=> []
=> []
=> ? = 0
[5,5,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 8
[5,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 8
[5,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 9
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 10
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 10
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 10
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 11
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000369
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 88%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 88%
Values
[1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[2]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[4]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,1]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[2,1,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[5]
=> [1,1,1,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,1]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2]
=> [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,1]
=> [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[2,2,1]
=> [3,2]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[6]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[4,2]
=> [2,2,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[3,3]
=> [2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,2,1]
=> [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3
[3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[2,2,2]
=> [3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 4
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[7]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[4,3]
=> [2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[4,2,1]
=> [3,2,1,1]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 3
[4,1,1,1]
=> [4,1,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[3,3,1]
=> [3,2,2]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[3,2,2]
=> [3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 4
[3,2,1,1]
=> [4,2,1]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 4
[2,2,2,1]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[2,2,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 5
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6
[8]
=> [1,1,1,1,1,1,1,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 3
[5,2,1]
=> [3,2,1,1,1]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[5,1,1,1]
=> [4,1,1,1,1]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 3
[3,2,2,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[2,2,2,2]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,2,2,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 5
[3,3,2,1]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[3,2,2,2]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[2,2,2,2,1]
=> [5,4]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[5,2,2,1]
=> [4,3,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5
[4,3,2,1]
=> [4,3,2,1]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6
[4,2,2,2]
=> [4,4,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 6
[3,3,2,2]
=> [4,4,2]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[3,3,2,1,1]
=> [5,3,2]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[3,2,2,2,1]
=> [5,4,1]
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 7
[2,2,2,2,2]
=> [5,5]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[2,2,1,1,1,1,1,1]
=> [8,2]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 8
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 9
[6,2,2,1]
=> [4,3,1,1,1,1]
=> [7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 5
[5,3,2,1]
=> [4,3,2,1,1]
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 6
[5,2,2,2]
=> [4,4,1,1,1]
=> [8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 6
[4,4,1,1,1]
=> [5,2,2,2]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 7
[4,3,2,2]
=> [4,4,2,1]
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 7
[4,3,2,1,1]
=> [5,3,2,1]
=> [7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 7
[4,2,2,2,1]
=> [5,4,1,1]
=> [9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 7
[3,3,3,1,1]
=> [5,3,3]
=> [8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[3,3,2,2,1]
=> [5,4,2]
=> [9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[3,3,2,1,1,1]
=> [6,3,2]
=> [7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
[3,2,2,2,2]
=> [5,5,1]
=> [10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 8
[3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 8
[2,2,2,2,2,1]
=> [6,5]
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[2,2,2,1,1,1,1,1]
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 9
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
[7,2,2,1]
=> [4,3,1,1,1,1,1]
=> [7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 5
[6,4,2]
=> [3,3,2,2,1,1]
=> [7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6
[6,2,2,2]
=> [4,4,1,1,1,1]
=> [8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 6
[5,5,2]
=> [3,3,2,2,2]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 7
[5,4,1,1,1]
=> [5,2,2,2,1]
=> [7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 7
[5,3,2,2]
=> [4,4,2,1,1]
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[5,3,2,1,1]
=> [5,3,2,1,1]
=> [7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
[5,2,2,2,1]
=> [5,4,1,1,1]
=> [9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[5,1,1,1,1,1,1,1]
=> [8,1,1,1,1]
=> [4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 7
[4,4,4]
=> [3,3,3,3]
=> [7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
The following 93 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000719The number of alignments in a perfect matching. St000377The dinv defect of an integer partition. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000738The first entry in the last row of a standard tableau. St000507The number of ascents of a standard tableau. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000734The last entry in the first row of a standard tableau. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000157The number of descents of a standard tableau. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. 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. St000662The staircase size of the code of a permutation. St000074The number of special entries. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000013The height of a Dyck path. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001726The number of visible inversions of a permutation. St000439The position of the first down step of a Dyck path. St000839The largest opener of a set partition. 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. St000030The sum of the descent differences of a permutations. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000505The biggest entry in the block containing the 1. St000019The cardinality of the support of a permutation. St000971The smallest closer of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St001809The index of the step at the first peak of maximal height in a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000728The dimension of a set partition. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000171The degree of the graph. St000211The rank of the set partition. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000025The number of initial rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001580The acyclic chromatic number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001277The degeneracy of a graph. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000223The number of nestings in the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000829The Ulam distance of a permutation to the identity permutation. St000653The last descent of a permutation. St000005The bounce statistic of a Dyck path. St001727The number of invisible inversions of a permutation. St001118The acyclic chromatic index of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000021The number of descents of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000325The width of the tree associated to 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000216The absolute length of a permutation. St001330The hat guessing number of a graph. St001894The depth of a signed permutation.
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