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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000475
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(load all 8 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2]
=> 0
[1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [3]
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,2]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [3,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [4,1]
=> 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000445
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5,8] => [2,1,4,3,7,6,5,8] => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5,8] => [2,1,7,6,4,3,5,8] => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5,8] => [7,6,2,1,4,3,5,8] => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5,8] => [7,2,1,6,4,3,5,8] => ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => [8,7,2,1,6,4,3,5] => ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => [5,4,2,1,3,7,6,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6,8] => [5,2,1,4,3,7,6,8] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6,8] => [7,5,2,1,4,3,6,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,7,3,8,6] => [8,7,5,4,2,1,3,6] => ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,7,3,8,6] => [8,7,5,2,1,4,3,6] => ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,5,7,8,1,3,6] => [8,2,1,7,5,4,3,6] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,4,8,3,5,6,7] => [2,1,8,4,3,5,6,7] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,8,1,3,5,6,7] => [8,2,1,4,3,5,6,7] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7] => [5,4,2,1,3,8,6,7] => ?
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => [5,4,2,1,3,6,8,7] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,6,8,1,3,7] => [8,2,1,6,5,4,3,7] => ?
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7,8] => [6,5,2,1,4,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,4,5,6,1,7,8,3] => [8,7,6,2,1,5,4,3] => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,8,7] => [6,2,1,3,5,4,8,7] => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,8,4,7] => [2,1,8,6,5,3,4,7] => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,8,3,4,7] => [2,1,8,3,6,5,4,7] => ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,6,8,1,3,4,7] => [8,2,1,6,3,5,4,7] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7,8] => [6,5,2,1,3,4,7,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7,8] => [2,1,6,3,5,4,7,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7,8] => [6,2,1,3,5,4,7,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4,8] => [2,1,7,6,3,5,4,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4,8] => [7,6,2,1,3,5,4,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,5,6,1,3,7,4,8] => [7,6,2,1,5,3,4,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4,8] => [2,1,7,3,6,5,4,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,5,1,6,3,7,8,4] => [8,7,6,2,1,3,5,4] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,6,7,1,3,8,4] => [8,7,2,1,6,3,5,4] => ?
=> ? = 0
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,5,6,7,8,1,3,4] => [8,2,1,7,3,6,5,4] => ?
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,8,4,6] => [8,5,2,1,3,4,7,6] => ?
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,7,8,3,4,6] => [8,2,1,3,7,5,4,6] => ?
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,7,3,4,6,8] => [2,1,7,3,5,4,6,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,7,6,8] => [5,3,2,1,4,7,6,8] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,5,7,8,1,4,6] => [8,3,2,1,7,5,4,6] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,8,3,4,6,7] => [2,1,8,3,5,4,6,7] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,5,1,8,3,4,6,7] => [8,2,1,3,5,4,6,7] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,5,8,1,3,4,6,7] => [8,2,1,5,3,4,6,7] => ?
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,5,4,6,7,8] => [3,2,1,5,4,6,7,8] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,5,6,7,1,4,8] => [7,3,2,1,6,5,4,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,6,1,3,7,8,4,5] => [8,2,1,3,4,7,6,5] => ?
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [2,6,7,1,8,3,4,5] => [8,2,1,7,3,4,6,5] => ?
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [8,2,1,7,3,6,4,5] => ?
=> ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [2,1,6,3,7,4,5,8] => [2,1,7,3,4,6,5,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [2,6,7,1,3,4,5,8] => [7,2,1,6,3,4,5,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [2,6,1,3,4,7,5,8] => [7,6,2,1,3,4,5,8] => ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [2,1,6,3,8,4,5,7] => [2,1,8,3,4,6,5,7] => ?
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [2,1,6,3,4,8,5,7] => [2,1,8,6,3,4,5,7] => ?
=> ? = 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000674
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000247
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => {{1}}
=> {{1}}
=> ? = 1
[1,0,1,0]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 0
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => {{1,2,3,4},{5,6},{7,8}}
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,8,7] => {{1,2,4},{3,5,6},{7,8}}
=> {{1,2,5,6},{3,4},{7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,8,5,7] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => {{1,2},{3,4,6},{5,7,8}}
=> {{1,2},{3,4,7,8},{5,6}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => {{1,2,4,5,7,8},{3,6}}
=> {{1,2,5,7,8},{3,4,6}}
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => {{1,2},{3,4},{5,6},{7},{8}}
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7,8] => {{1,2,3,4},{5,6},{7},{8}}
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7,8] => {{1,2,4},{3,5,6},{7},{8}}
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5,8] => {{1,2},{3,4},{5,6,7},{8}}
=> {{1,2},{3,4},{5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5,8] => {{1,2},{3,4,5,6,7},{8}}
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5,8] => {{1,2,4},{3,5,6,7},{8}}
=> {{1,2,5,6,7},{3,4},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5,8] => {{1,2,4,5,7},{3,6},{8}}
=> {{1,2,5,7},{3,4,6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,7,8,5] => {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,8,5] => {{1,2},{3,4,6},{5,7,8}}
=> {{1,2},{3,4,7,8},{5,6}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,6,1,7,3,8,5] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => {{1,2,4,5,7,8},{3,6}}
=> {{1,2,5,7,8},{3,4,6}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,6,7,8,1,3,5] => {{1,2,3,4,6,7},{5,8}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,8,6] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,8,6] => {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,7,8,5,6] => {{1,2},{3,4},{5,7},{6,8}}
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,7,8,5,6] => {{1,2,3,4},{5,7},{6,8}}
=> {{1,2,3,4},{5,8},{6,7}}
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => {{1,2,3,4,5},{6,7},{8}}
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6,8] => {{1,2,4},{3,5},{6,7},{8}}
=> {{1,2,5},{3,4},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6,8] => {{1,2,4},{3,5,6,7},{8}}
=> {{1,2,5,6,7},{3,4},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => {{1,2},{3,4},{5},{6,7,8}}
=> {{1,2},{3,4},{5},{6,7,8}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,7,8,6] => {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,7,8,6] => {{1,2,4},{3,5},{6,7,8}}
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,7,8,3,6] => {{1,2},{3,4,5,7},{6,8}}
=> {{1,2},{3,4,5,8},{6,7}}
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,7,8,3,6] => {{1,2,4},{3,5,7},{6,8}}
=> {{1,2,5,8},{3,4},{6,7}}
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,4,3,8,5,6,7] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,1,3,8,5,6,7] => {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,8,6,7] => {{1,2},{3,4},{5},{6,7,8}}
=> {{1,2},{3,4},{5},{6,7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7] => {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,8,7] => {{1,2},{3,4},{5},{6},{7,8}}
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,8,7] => {{1,2},{3,4,5},{6},{7,8}}
=> {{1,2},{3,4,5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => {{1,2,3,4,5},{6},{7,8}}
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,8,7] => {{1,2,4},{3,5,6},{7,8}}
=> {{1,2,5,6},{3,4},{7,8}}
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,6,8,1,3,7] => {{1,2,4,6},{3,5,7,8}}
=> {{1,2,7,8},{3,4,5,6}}
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8] => {{1,2},{3,4},{5},{6},{7},{8}}
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7,8] => {{1,2,3,4},{5},{6},{7},{8}}
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7,8] => {{1,2},{3,4,5},{6},{7},{8}}
=> {{1,2},{3,4,5},{6},{7},{8}}
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7,8] => {{1,2,4},{3,5},{6},{7},{8}}
=> {{1,2,5},{3,4},{6},{7},{8}}
=> ? = 3
Description
The number of singleton blocks of a set partition.
Matching statistic: St000022
(load all 72 compositions to match this statistic)
(load all 72 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 92%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1] => {{1}}
=> [1] => 1
[1,0,1,0]
=> [2,1] => {{1,2}}
=> [2,1] => 0
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> [1,2] => 2
[1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> [2,3,1] => 0
[1,1,0,0,1,0]
=> [3,1,2] => {{1,2,3}}
=> [2,3,1] => 0
[1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 2
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => {{1,2},{3,4},{5,6},{7}}
=> [2,1,4,3,6,5,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => {{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => {{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => {{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => {{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => {{1,2},{3,4,6},{5,7}}
=> [2,1,4,6,7,3,5] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => {{1,2,4},{3,6},{5,7}}
=> [2,4,6,1,7,3,5] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => {{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => {{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => {{1,2},{3,4},{5},{6,7}}
=> [2,1,4,3,5,7,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => {{1,2},{3,4,5},{6,7}}
=> [2,1,4,5,3,7,6] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => {{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => {{1,2},{3,4},{5},{6},{7}}
=> [2,1,4,3,5,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => {{1,2},{3,4,5},{6},{7}}
=> [2,1,4,5,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => {{1,2,4},{3,5},{6},{7}}
=> [2,4,5,1,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => {{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => {{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => {{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => {{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => {{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => {{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => {{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,5,6,1,7,3,4] => {{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => ? = 0
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => {{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,7,4,6] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,7,3,4,6] => {{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,7,6] => {{1,2},{3,4,5},{6,7}}
=> [2,1,4,5,3,7,6] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => {{1,2},{3},{4,5},{6,7}}
=> [2,1,3,5,4,7,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => {{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6] => {{1,2},{3},{4,5,6,7}}
=> [2,1,3,5,6,7,4] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,4,6] => {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => {{1,2},{3,4,5},{6},{7}}
=> [2,1,4,5,3,6,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,3,5,4,6,7] => {{1,2},{3},{4,5},{6},{7}}
=> [2,1,3,5,4,6,7] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,6,7] => {{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,3,5,6,4,7] => {{1,2},{3},{4,5,6},{7}}
=> [2,1,3,5,6,4,7] => ? = 2
Description
The number of fixed points of a permutation.
Matching statistic: St000160
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 92%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [3,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [4,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [3,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [4,1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [2,2,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [4,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [3,2,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [5,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [3,2,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [3,2,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [5,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => [2,2,1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => [3,2,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => [5,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => [2,1,1,1,1]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => [3,1,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,5,2,6] => [4,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => [5,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => [4,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,6,3] => [5,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => [3,2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => [5,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => [5,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => [3,2,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => [3,2,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => [4,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => [3,1,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => [2,1,1,1,1]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,5,3,6] => [3,1,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => [4,1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,2,5,6,8,4,7] => ?
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,5,6,7,2,8,4] => ?
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,3,6,7,2,4,5,8] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5,8,7] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,3,4,2,6,8,5,7] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,3,4,6,2,8,5,7] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,3,2,7,8,4,5,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,3,4,2,7,5,6,8] => ?
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,3,4,5,7,2,6,8] => ?
=> ? = 1 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,4,5,7,2,8,3,6] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,8,3,6,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,5,6,3,8,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,5,6,8,3,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,4,2,5,6,3,7,8] => ?
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,4,5,6,2,7,8,3] => ?
=> ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,4,6,2,3,7,5,8] => ?
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,7,8,5] => ?
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,6,8,5,7] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? = 2 + 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ? = 1 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 1 + 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,4,5,7,3,8,6] => ?
=> ? = 1 + 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2 + 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000248
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 75%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => {{1}}
=> {{1}}
=> ? = 1
[1,0,1,0]
=> [2,1] => {{1,2}}
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> {{1,2}}
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => {{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,2,3,5},{4}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => {{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => {{1,2,3,4},{5,6},{7,8}}
=> {{1,3,5},{2},{4},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> {{1,3,7},{2},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => {{1,2,3,4,5,6},{7,8}}
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,8,7] => {{1,2,4},{3,5,6},{7,8}}
=> {{1,3,6},{2},{4},{5,7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,8,5,7] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => {{1,2,3,4},{5,6,7,8}}
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,8,5,7] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,8,5,7] => {{1,2,4},{3,5,6,7,8}}
=> {{1,6},{2},{3},{4},{5,7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => {{1,2},{3,4,6},{5,7,8}}
=> {{1,4,7},{2},{3,5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,8,3,5,7] => {{1,2,3,4,6},{5,7,8}}
=> {{1,4},{2},{3,5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,4},{2},{3,6},{5,7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => {{1,2,4,5,7,8},{3,6}}
=> {{1},{2},{3,6},{4,7},{5},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => {{1,2},{3,4},{5,6},{7},{8}}
=> {{1,2,3,5,7},{4},{6},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7,8] => {{1,2,3,4},{5,6},{7},{8}}
=> {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7,8] => {{1,2,4},{3,5,6},{7},{8}}
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5,8] => {{1,2},{3,4},{5,6,7},{8}}
=> {{1,2,5,7},{3},{4},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5,8] => {{1,2},{3,4,5,6,7},{8}}
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5,8] => {{1,2,3,4,5,6,7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5,8] => {{1,2,4},{3,5,6,7},{8}}
=> {{1,2,6},{3},{4},{5,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5,8] => {{1,2,4,5,7},{3,6},{8}}
=> {{1,2},{3,6},{4,7},{5},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,7,8,5] => {{1,2,3,4},{5,6,7,8}}
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,3,7,8,5] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,8,5] => {{1,2},{3,4,6},{5,7,8}}
=> {{1,4,7},{2},{3,5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,6,7,3,8,5] => {{1,2,3,4,6},{5,7,8}}
=> {{1,4},{2},{3,5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,6,1,7,3,8,5] => {{1,2,4},{3,6},{5,7,8}}
=> {{1,4},{2},{3,6},{5,7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => {{1,2,4,5,7,8},{3,6}}
=> {{1},{2},{3,6},{4,7},{5},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,6,7,8,3,5] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,6,7,8,3,5] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,6,7,8,1,3,5] => {{1,2,3,4,6,7},{5,8}}
=> {{1,4},{2,5},{3},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,8,6] => {{1,2},{3,4},{5,6,7,8}}
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,8,6] => {{1,2,3,4},{5,6,7,8}}
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,8,6] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,8,6] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,7,8,5,6] => {{1,2},{3,4},{5,7},{6,8}}
=> {{1,3,5,7},{2,4},{6},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,7,8,5,6] => {{1,2,3,4},{5,7},{6,8}}
=> {{1,3,5},{2,4},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,4,1,7,3,8,5,6] => {{1,2,3,4,5,7},{6,8}}
=> {{1,3},{2,4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,7,8,3,5,6] => {{1,2},{3,4,5,6,7,8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,8,3,5,6] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4,7,1,8,3,5,6] => {{1,2,4},{3,5,6,7,8}}
=> {{1,6},{2},{3},{4},{5,7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => {{1,2,3,4,5},{6,7},{8}}
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6,8] => {{1,2,4},{3,5},{6,7},{8}}
=> {{1,2,4,6},{3},{5,7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6,8] => {{1,2,4},{3,5,6,7},{8}}
=> {{1,2,6},{3},{4},{5,7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => {{1,2},{3,4},{5},{6,7,8}}
=> {{1,4,5,7},{2},{3},{6},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,7,8,6] => {{1,2,3,4,5},{6,7,8}}
=> {{1,4},{2},{3},{5},{6},{7},{8}}
=> ? = 0
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 92%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000215
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 92%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,2,3,1,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,4,3,1,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,3,2,1,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,2,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,1,4,2] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,2,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,3,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,3,2,5,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [4,3,6,1,5,2,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => [5,2,4,3,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [6,1,5,2,4,3,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [5,2,6,1,4,3,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,1,7] => [3,4,5,2,6,1,7] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => [3,4,6,1,5,2,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,1,7] => [3,5,2,4,6,1,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [3,6,1,5,2,4,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => [3,5,2,6,1,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,1,7] => [5,2,3,4,6,1,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [6,1,5,2,3,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => [5,2,3,6,1,4,7] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => [6,1,3,5,2,4,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,1,7] => [5,3,4,2,6,1,7] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,1,7] => [6,1,5,3,4,2,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,1,7] => [4,2,5,3,6,1,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [4,2,6,1,5,3,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,1,7] => [6,1,4,2,5,3,7] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,1,7] => [3,5,4,2,6,1,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => [3,6,1,5,4,2,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,1,7] => [3,4,2,5,6,1,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,1,7] => [3,4,2,6,1,5,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => [3,6,1,4,2,5,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,1,7] => [4,2,3,5,6,1,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [4,2,3,6,1,5,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => [6,1,4,2,3,5,7] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [4,2,6,1,3,5,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [6,1,5,4,3,2,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,5,1,7] => [4,3,2,5,6,1,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [4,3,2,6,1,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => [6,1,4,3,2,5,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,1,7] => [3,2,4,5,6,1,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [3,2,4,6,1,5,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [3,2,6,1,4,5,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [6,1,3,2,4,5,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => [2,3,4,6,1,5,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [2,3,6,1,4,5,7] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => [4,3,6,2,5,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,1,2,7] => [4,3,5,1,6,2,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,1,2,7] => [6,2,4,3,5,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,1,2,7] => [5,1,6,2,4,3,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,1,2,7] => [6,2,5,1,4,3,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,1,2,7] => [3,4,6,2,5,1,7] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,1,2,7] => [3,4,5,1,6,2,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,1,2,7] => [3,6,2,4,5,1,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,1,2,7] => [3,5,1,6,2,4,7] => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,1,2,7] => [3,6,2,5,1,4,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,1,2,7] => [6,2,3,4,5,1,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,1,2,7] => [6,2,3,5,1,4,7] => ? = 0
Description
The number of adjacencies of a permutation, zero appended.
An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended.
$(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000895
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 58%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1,0]
=> [[1]]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,-1,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,-1,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,-1,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
Description
The number of ones on the main diagonal of an alternating sign matrix.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001479The number of bridges of a graph. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000315The number of isolated vertices of a graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000153The number of adjacent cycles of a permutation. St001545The second Elser number of a connected graph. St000894The trace of an alternating sign matrix. St000907The number of maximal antichains of minimal length in a poset. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001903The number of fixed points of a parking function.
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