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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001688
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(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 4
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 5
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 9
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 6
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 8
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 10
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 16
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 7
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 9
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 11
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 13
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 17
[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]
=> [1,1,1,1,1,0,0,0,0,0]
=> 25
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 8
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 10
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 12
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 12
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 14
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 18
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 18
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 20
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 26
[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]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 36
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 9
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 11
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 13
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 13
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 15
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 19
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 17
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 19
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 21
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 27
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 25
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 29
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 37
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 49
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 12
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 14
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 14
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 16
Description
The sum of the squares of the heights of the peaks of a Dyck path.
Matching statistic: St001127
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001127: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 83%●distinct values known / distinct values provided: 82%
St001127: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 83%●distinct values known / distinct values provided: 82%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 2
[1,1]
=> [2]
=> 4
[3]
=> [1,1,1]
=> 3
[2,1]
=> [2,1]
=> 5
[1,1,1]
=> [3]
=> 9
[4]
=> [1,1,1,1]
=> 4
[3,1]
=> [2,1,1]
=> 6
[2,2]
=> [2,2]
=> 8
[2,1,1]
=> [3,1]
=> 10
[1,1,1,1]
=> [4]
=> 16
[5]
=> [1,1,1,1,1]
=> 5
[4,1]
=> [2,1,1,1]
=> 7
[3,2]
=> [2,2,1]
=> 9
[3,1,1]
=> [3,1,1]
=> 11
[2,2,1]
=> [3,2]
=> 13
[2,1,1,1]
=> [4,1]
=> 17
[1,1,1,1,1]
=> [5]
=> 25
[6]
=> [1,1,1,1,1,1]
=> 6
[5,1]
=> [2,1,1,1,1]
=> 8
[4,2]
=> [2,2,1,1]
=> 10
[4,1,1]
=> [3,1,1,1]
=> 12
[3,3]
=> [2,2,2]
=> 12
[3,2,1]
=> [3,2,1]
=> 14
[3,1,1,1]
=> [4,1,1]
=> 18
[2,2,2]
=> [3,3]
=> 18
[2,2,1,1]
=> [4,2]
=> 20
[2,1,1,1,1]
=> [5,1]
=> 26
[1,1,1,1,1,1]
=> [6]
=> 36
[7]
=> [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [2,1,1,1,1,1]
=> 9
[5,2]
=> [2,2,1,1,1]
=> 11
[5,1,1]
=> [3,1,1,1,1]
=> 13
[4,3]
=> [2,2,2,1]
=> 13
[4,2,1]
=> [3,2,1,1]
=> 15
[4,1,1,1]
=> [4,1,1,1]
=> 19
[3,3,1]
=> [3,2,2]
=> 17
[3,2,2]
=> [3,3,1]
=> 19
[3,2,1,1]
=> [4,2,1]
=> 21
[3,1,1,1,1]
=> [5,1,1]
=> 27
[2,2,2,1]
=> [4,3]
=> 25
[2,2,1,1,1]
=> [5,2]
=> 29
[2,1,1,1,1,1]
=> [6,1]
=> 37
[1,1,1,1,1,1,1]
=> [7]
=> 49
[8]
=> [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [2,1,1,1,1,1,1]
=> 10
[6,2]
=> [2,2,1,1,1,1]
=> 12
[6,1,1]
=> [3,1,1,1,1,1]
=> 14
[5,3]
=> [2,2,2,1,1]
=> 14
[5,2,1]
=> [3,2,1,1,1]
=> 16
[7,6]
=> [2,2,2,2,2,2,1]
=> ? = 25
[6,5,2]
=> [3,3,2,2,2,1]
=> ? = 31
[6,4,3]
=> [3,3,3,2,1,1]
=> ? = 33
[5,5,3]
=> [3,3,3,2,2]
=> ? = 35
[5,4,4]
=> [3,3,3,3,1]
=> ? = 37
[4,4,4,1]
=> [4,3,3,3]
=> ? = 43
[4,3,3,3]
=> [4,4,4,1]
=> ? = 49
[4,3,2,2,2]
=> [5,5,2,1]
=> ? = 55
[3,3,3,3,1]
=> [5,4,4]
=> ? = 57
[3,3,3,2,2]
=> [5,5,3]
=> ? = 59
[3,2,2,2,2,2]
=> [6,6,1]
=> ? = 73
[6,5,3]
=> [3,3,3,2,2,1]
=> ? = 36
[6,4,4]
=> [3,3,3,3,1,1]
=> ? = 38
[5,5,4]
=> [3,3,3,3,2]
=> ? = 40
[5,4,4,1]
=> [4,3,3,3,1]
=> ? = 44
[5,3,3,3]
=> [4,4,4,1,1]
=> ? = 50
[4,4,4,2]
=> [4,4,3,3]
=> ? = 50
[4,4,3,3]
=> [4,4,4,2]
=> ? = 52
[4,3,3,3,1]
=> [5,4,4,1]
=> ? = 58
[4,3,3,2,2]
=> [5,5,3,1]
=> ? = 60
[3,3,3,3,2]
=> [5,5,4]
=> ? = 66
[6,5,4]
=> [3,3,3,3,2,1]
=> ? = 41
[5,5,5]
=> [3,3,3,3,3]
=> ? = 45
[5,4,4,2]
=> [4,4,3,3,1]
=> ? = 51
[5,4,3,3]
=> [4,4,4,2,1]
=> ? = 53
[4,4,4,3]
=> [4,4,4,3]
=> ? = 57
[4,3,3,3,2]
=> [5,5,4,1]
=> ? = 67
[3,3,3,3,3]
=> [5,5,5]
=> ? = 75
[6,5,5]
=> [3,3,3,3,3,1]
=> ? = 46
[5,4,4,3]
=> [4,4,4,3,1]
=> ? = 58
[4,4,4,4]
=> [4,4,4,4]
=> ? = 64
[4,3,3,3,3]
=> [5,5,5,1]
=> ? = 76
[5,4,4,4]
=> [4,4,4,4,1]
=> ? = 65
Description
The sum of the squares of the parts of a partition.
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