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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000011
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 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: St001462
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [1] => [[1]]
=> 1
{{1,2}}
=> [2,1] => [2,1] => [[1],[2]]
=> 1
{{1},{2}}
=> [1,2] => [1,2] => [[1,2]]
=> 2
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 5
{{1},{2},{3,4},{5,6,7,8}}
=> [1,2,4,3,6,7,8,5] => [1,2,4,3,8,6,7,5] => ?
=> ? = 4
{{1},{2},{3,4,5,6,8},{7}}
=> [1,2,4,5,6,8,7,3] => [1,2,8,4,5,7,6,3] => ?
=> ? = 3
{{1},{2,4,5},{3},{6,7},{8}}
=> [1,4,3,5,2,7,6,8] => [1,5,3,4,2,7,6,8] => ?
=> ? = 4
{{1},{2,3,4,5,6,8},{7}}
=> [1,3,4,5,6,8,7,2] => [1,8,3,4,5,7,6,2] => ?
=> ? = 2
{{1,4,7,8},{2},{3},{5,6}}
=> [4,2,3,7,6,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ? = 1
{{1,4,5,6,7},{2},{3},{8}}
=> [4,2,3,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ? = 2
{{1,4,5,6,8},{2},{3},{7}}
=> [4,2,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? = 1
{{1,5,6,7},{2},{3,4},{8}}
=> [5,2,4,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 2
{{1,3,5,6,8},{2},{4},{7}}
=> [3,2,5,4,6,8,7,1] => [8,2,4,3,5,7,6,1] => ?
=> ? = 1
{{1,3,5,6,7,8},{2},{4}}
=> [3,2,5,4,6,7,8,1] => [8,2,4,3,5,6,7,1] => ?
=> ? = 1
{{1,3,4,5,7,8},{2},{6}}
=> [3,2,4,5,7,6,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ? = 1
{{1,2,3},{4,5},{6},{7},{8}}
=> [2,3,1,5,4,6,7,8] => [3,2,1,5,4,6,7,8] => ?
=> ? = 5
{{1,2,3},{4,6,7},{5},{8}}
=> [2,3,1,6,5,7,4,8] => [3,2,1,7,5,6,4,8] => ?
=> ? = 3
{{1,2,3},{4,5,6},{7},{8}}
=> [2,3,1,5,6,4,7,8] => [3,2,1,6,5,4,7,8] => ?
=> ? = 4
{{1,4,7,8},{2,3},{5},{6}}
=> [4,3,2,7,5,6,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ? = 1
{{1,4,6,7,8},{2,3},{5}}
=> [4,3,2,6,5,7,8,1] => [8,3,2,5,4,6,7,1] => ?
=> ? = 1
{{1,4,5,6,7},{2,3},{8}}
=> [4,3,2,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ? = 2
{{1,4,5,6,8},{2,3},{7}}
=> [4,3,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? = 1
{{1,5,6,7},{2,4},{3},{8}}
=> [5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 2
{{1,2,4,7,8},{3},{5},{6}}
=> [2,4,3,7,5,6,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ? = 1
{{1,2,4,6,7,8},{3},{5}}
=> [2,4,3,6,5,7,8,1] => [8,3,2,5,4,6,7,1] => ?
=> ? = 1
{{1,2,4,7,8},{3},{5,6}}
=> [2,4,3,7,6,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ? = 1
{{1,2,4,5,6,8},{3},{7}}
=> [2,4,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? = 1
{{1,2,3,4},{5,6},{7},{8}}
=> [2,3,4,1,6,5,7,8] => [4,2,3,1,6,5,7,8] => ?
=> ? = 4
{{1,2,3,4},{5,6,7},{8}}
=> [2,3,4,1,6,7,5,8] => [4,2,3,1,7,6,5,8] => ?
=> ? = 3
{{1,2,3,5,6,7,8},{4}}
=> [2,3,5,4,6,7,8,1] => [8,2,4,3,5,6,7,1] => ?
=> ? = 1
{{1,2,3,4,5},{6,7},{8}}
=> [2,3,4,5,1,7,6,8] => [5,2,3,4,1,7,6,8] => ?
=> ? = 3
{{1,2,3,6},{4,5},{7},{8}}
=> [2,3,6,5,4,1,7,8] => [6,2,5,4,3,1,7,8] => ?
=> ? = 3
{{1,2,3,4,5,7,8},{6}}
=> [2,3,4,5,7,6,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ? = 1
{{1,2,3,4,6,7},{5,8}}
=> [2,3,4,6,8,7,1,5] => [7,2,3,8,6,5,1,4] => ?
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9]]
=> ? = 9
{{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 8
{{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,9,8,7] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 7
{{1},{2},{3,9},{4},{5},{6},{7},{8}}
=> [1,2,9,4,5,6,7,8,3] => [1,2,9,8,5,6,7,4,3] => ?
=> ? = 3
{{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> [1,9,3,4,5,6,7,8,2] => [1,9,8,4,5,6,7,3,2] => [[1,2,5,6,7],[3],[4],[8],[9]]
=> ? = 2
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [9,2,3,4,5,6,7,8,1] => [9,8,3,4,5,6,7,2,1] => ?
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 9
{{1},{2},{3},{4},{5},{6},{7,8,9}}
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,9,8,7] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 7
{{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
=> [1,2,10,4,5,6,7,8,9,3] => [1,2,10,9,5,6,7,8,4,3] => ?
=> ? = 3
{{1,9},{2,3},{4},{5},{6},{7},{8}}
=> [9,3,2,4,5,6,7,8,1] => [9,8,3,4,5,6,7,2,1] => ?
=> ? = 1
{{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
=> [1,10,3,4,5,6,7,8,9,2] => [1,10,9,4,5,6,7,8,3,2] => [[1,2,5,6,7,8],[3],[4],[9],[10]]
=> ? = 2
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [10,2,3,4,5,6,7,8,9,1] => [10,9,3,4,5,6,7,8,2,1] => ?
=> ? = 1
{{1,2,3,4,5,6,7,8},{9}}
=> [2,3,4,5,6,7,8,1,9] => [8,2,3,4,5,6,7,1,9] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? = 2
{{1},{2,3,4,5,6,7,8,9}}
=> [1,3,4,5,6,7,8,9,2] => [1,9,3,4,5,6,7,8,2] => [[1,2,4,5,6,7,8],[3],[9]]
=> ? = 2
{{1,2,3,4,5,6,7,8,9},{10}}
=> [2,3,4,5,6,7,8,9,1,10] => [9,2,3,4,5,6,7,8,1,10] => [[1,3,4,5,6,7,8,10],[2],[9]]
=> ? = 2
{{1,2,3,4,5,6,8},{7},{9}}
=> [2,3,4,5,6,8,7,1,9] => [8,2,3,4,5,7,6,1,9] => ?
=> ? = 2
{{1},{2,3,4,5,6,7,9},{8}}
=> [1,3,4,5,6,7,9,8,2] => [1,9,3,4,5,6,8,7,2] => ?
=> ? = 2
{{1},{2,3,4,5,6,7,8,9,10}}
=> [1,3,4,5,6,7,8,9,10,2] => [1,10,3,4,5,6,7,8,9,2] => [[1,2,4,5,6,7,8,9],[3],[10]]
=> ? = 2
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St000439
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> 2 = 1 + 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
{{1,2},{3,5},{4,6},{7,8}}
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,6},{4,5},{7,8}}
=> [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,7},{4,5},{6,8}}
=> [2,1,7,5,4,8,3,6] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,7},{4,6},{5,8}}
=> [2,1,7,6,8,4,3,5] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,6},{4,7},{5,8}}
=> [2,1,6,7,8,3,4,5] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,5},{4,7},{6,8}}
=> [2,1,5,7,3,8,4,6] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,3},{2,4},{5,7},{6,8}}
=> [3,4,1,2,7,8,5,6] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
{{1,2},{3,4},{5,7},{6,8}}
=> [2,1,4,3,7,8,5,6] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,4},{5,8},{6,7}}
=> [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
{{1,2},{3,5},{4,8},{6,7}}
=> [2,1,5,8,3,7,6,4] => [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,6},{4,8},{5,7}}
=> [2,1,6,8,7,3,5,4] => [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,7},{4,8},{5,6}}
=> [2,1,7,8,6,5,3,4] => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
{{1},{2},{3},{4,7,8},{5,6}}
=> [1,2,3,7,6,5,8,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
{{1},{2},{3},{4,8},{5,6,7}}
=> [1,2,3,8,6,7,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
{{1},{2},{3,6,7,8},{4,5}}
=> [1,2,6,5,4,7,8,3] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1},{2},{3,7,8},{4,6},{5}}
=> [1,2,7,6,5,4,8,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1},{2,3},{4,5},{6,7},{8}}
=> [1,3,2,5,4,7,6,8] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 5 + 1
{{1},{2,4},{3},{5},{6,7,8}}
=> [1,4,3,2,5,7,8,6] => [1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
{{1},{2,4},{3},{5,8},{6,7}}
=> [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
{{1},{2,5,6,7,8},{3},{4}}
=> [1,5,3,4,6,7,8,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
{{1},{2,4,5},{3},{6,7},{8}}
=> [1,4,3,5,2,7,6,8] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 + 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [1,4,3,6,5,7,2,8] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
{{1},{2,4,5,6,7,8},{3}}
=> [1,4,3,5,6,7,8,2] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
{{1},{2,3,4},{5},{6,7,8}}
=> [1,3,4,2,5,7,8,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
{{1},{2,3,4},{5,8},{6,7}}
=> [1,3,4,2,8,7,6,5] => [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
{{1},{2,5,6,7,8},{3,4}}
=> [1,5,4,3,6,7,8,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
{{1},{2,6,7,8},{3,5},{4}}
=> [1,6,5,4,3,7,8,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
{{1,2},{3,4},{5},{6},{7},{8}}
=> [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 6 + 1
{{1,2},{3,5},{4},{6},{7},{8}}
=> [2,1,5,4,3,6,7,8] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 5 + 1
{{1,2},{3,7,8},{4},{5,6}}
=> [2,1,7,4,6,5,8,3] => [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,5,6,7,8},{4}}
=> [2,1,5,4,6,7,8,3] => [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,5},{6},{7},{8}}
=> [2,1,4,5,3,6,7,8] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 5 + 1
{{1,2},{3,7,8},{4,6},{5}}
=> [2,1,7,6,5,4,8,3] => [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,8},{5},{6,7}}
=> [2,1,4,8,5,7,6,3] => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,8},{5,6},{7}}
=> [2,1,4,8,6,5,7,3] => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,8},{5,7},{6}}
=> [2,1,4,8,7,6,5,3] => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,5,8},{6},{7}}
=> [2,1,4,5,8,6,7,3] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,8},{5,6,7}}
=> [2,1,4,8,6,7,5,3] => [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,5,8},{6,7}}
=> [2,1,4,5,8,7,6,3] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2},{3,4,5,6,7,8}}
=> [2,1,4,5,6,7,8,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,3},{2},{4},{5},{6},{7},{8}}
=> [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
{{1,3},{2},{4},{5},{6},{7,8}}
=> [3,2,1,4,5,6,8,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 + 1
{{1,3},{2},{4},{5},{6,7,8}}
=> [3,2,1,4,5,7,8,6] => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
{{1,4},{2},{3},{5},{6},{7},{8}}
=> [4,2,3,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1,4},{2},{3},{5,6},{7,8}}
=> [4,2,3,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
{{1,5},{2},{3},{4},{6},{7},{8}}
=> [5,2,3,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
{{1,3,4},{2},{5},{6},{7},{8}}
=> [3,2,4,1,5,6,7,8] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000288
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => => ? = 1 - 1
{{1,2}}
=> [2,1] => 0 => 0 = 1 - 1
{{1},{2}}
=> [1,2] => 1 => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => 00 => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => 01 => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => 00 => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => 10 => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => 11 => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => 000 => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => 001 => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => 000 => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => 010 => 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 011 => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => 000 => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => 000 => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 001 => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => 000 => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => 100 => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => 101 => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 000 => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 100 => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => 110 => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 111 => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 0000 => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0001 => 1 = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0000 => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0010 => 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0011 => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0000 => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0000 => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0001 => 1 = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 0000 => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0100 => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0101 => 2 = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0000 => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 0100 => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0110 => 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0111 => 3 = 4 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0000 => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 0000 => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0001 => 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 0000 => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0000 => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 0001 => 1 = 2 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0000 => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 0000 => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0010 => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0011 => 2 = 3 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 0000 => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0000 => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 0001 => 1 = 2 - 1
{{1},{2},{3,8},{4},{5,6,7}}
=> [1,2,8,4,6,7,5,3] => ? => ? = 3 - 1
{{1},{2},{3,7,8},{4,6},{5}}
=> [1,2,7,6,5,4,8,3] => ? => ? = 3 - 1
{{1},{2},{3,8},{4,5,6,7}}
=> [1,2,8,5,6,7,4,3] => ? => ? = 3 - 1
{{1},{2,8},{3,5},{4},{6,7}}
=> [1,8,5,4,3,7,6,2] => ? => ? = 2 - 1
{{1},{2,6,7,8},{3,5},{4}}
=> [1,6,5,4,3,7,8,2] => ? => ? = 2 - 1
{{1,2},{3,7,8},{4},{5,6}}
=> [2,1,7,4,6,5,8,3] => ? => ? = 2 - 1
{{1,2},{3,4,8},{5},{6,7}}
=> [2,1,4,8,5,7,6,3] => ? => ? = 2 - 1
{{1,2},{3,8},{4,5,6},{7}}
=> [2,1,8,5,6,4,7,3] => ? => ? = 2 - 1
{{1,2},{3,4,8},{5,6,7}}
=> [2,1,4,8,6,7,5,3] => ? => ? = 2 - 1
{{1,4},{2},{3},{5,6},{7,8}}
=> [4,2,3,1,6,5,8,7] => ? => ? = 3 - 1
{{1,6},{2},{3},{4},{5},{7,8}}
=> [6,2,3,4,5,1,8,7] => ? => ? = 2 - 1
{{1,8},{2},{3},{4},{5},{6,7}}
=> [8,2,3,4,5,7,6,1] => ? => ? = 1 - 1
{{1,8},{2},{3},{4},{5,7},{6}}
=> [8,2,3,4,7,6,5,1] => ? => ? = 1 - 1
{{1,6,7,8},{2},{3},{4,5}}
=> [6,2,3,5,4,7,8,1] => ? => ? = 1 - 1
{{1,4,8},{2},{3},{5,6,7}}
=> [4,2,3,8,6,7,5,1] => ? => ? = 1 - 1
{{1,4,5,8},{2},{3},{6,7}}
=> [4,2,3,5,8,7,6,1] => ? => ? = 1 - 1
{{1,5,8},{2},{3,4},{6,7}}
=> [5,2,4,3,8,7,6,1] => ? => ? = 1 - 1
{{1,6,7,8},{2},{3,5},{4}}
=> [6,2,5,4,3,7,8,1] => ? => ? = 1 - 1
{{1,7,8},{2},{3,5,6},{4}}
=> [7,2,5,4,6,3,8,1] => ? => ? = 1 - 1
{{1,3,5,6,8},{2},{4},{7}}
=> [3,2,5,4,6,8,7,1] => ? => ? = 1 - 1
{{1,6},{2},{3,4,5},{7,8}}
=> [6,2,4,5,3,1,8,7] => ? => ? = 2 - 1
{{1,6,8},{2},{3,4,5},{7}}
=> [6,2,4,5,3,8,7,1] => ? => ? = 1 - 1
{{1,7},{2},{3,6},{4,5},{8}}
=> [7,2,6,5,4,3,1,8] => ? => ? = 2 - 1
{{1,7,8},{2},{3,6},{4,5}}
=> [7,2,6,5,4,3,8,1] => ? => ? = 1 - 1
{{1,7,8},{2},{3,4,6},{5}}
=> [7,2,4,6,5,3,8,1] => ? => ? = 1 - 1
{{1,3,4,7,8},{2},{5},{6}}
=> [3,2,4,7,5,6,8,1] => ? => ? = 1 - 1
{{1,3,8},{2},{4,6,7},{5}}
=> [3,2,8,6,5,7,4,1] => ? => ? = 1 - 1
{{1,3,4,6,8},{2},{5},{7}}
=> [3,2,4,6,5,8,7,1] => ? => ? = 1 - 1
{{1,8},{2},{3,4,5,6},{7}}
=> [8,2,4,5,6,3,7,1] => ? => ? = 1 - 1
{{1,3,8},{2},{4,5,6},{7}}
=> [3,2,8,5,6,4,7,1] => ? => ? = 1 - 1
{{1,8},{2},{3,4,5,7},{6}}
=> [8,2,4,5,7,6,3,1] => ? => ? = 1 - 1
{{1,8},{2},{3,4,5,6,7}}
=> [8,2,4,5,6,7,3,1] => ? => ? = 1 - 1
{{1,3,4,5,6,8},{2},{7}}
=> [3,2,4,5,6,8,7,1] => ? => ? = 1 - 1
{{1,6,8},{2,3},{4,5},{7}}
=> [6,3,2,5,4,8,7,1] => ? => ? = 1 - 1
{{1,8},{2,3},{4,5,7},{6}}
=> [8,3,2,5,7,6,4,1] => ? => ? = 1 - 1
{{1,2,4},{3},{5,6},{7,8}}
=> [2,4,3,1,6,5,8,7] => ? => ? = 3 - 1
{{1,8},{2,4},{3},{5},{6,7}}
=> [8,4,3,2,5,7,6,1] => ? => ? = 1 - 1
{{1,7},{2,4},{3},{5,6},{8}}
=> [7,4,3,2,6,5,1,8] => ? => ? = 2 - 1
{{1,8},{2,4},{3},{5,7},{6}}
=> [8,4,3,2,7,6,5,1] => ? => ? = 1 - 1
{{1,8},{2,4},{3},{5,6,7}}
=> [8,4,3,2,6,7,5,1] => ? => ? = 1 - 1
{{1,6,7,8},{2,5},{3},{4}}
=> [6,5,3,4,2,7,8,1] => ? => ? = 1 - 1
{{1,7,8},{2,5,6},{3},{4}}
=> [7,5,3,4,6,2,8,1] => ? => ? = 1 - 1
{{1,2,5,7,8},{3},{4},{6}}
=> [2,5,3,4,7,6,8,1] => ? => ? = 1 - 1
{{1,2,5,6,8},{3},{4},{7}}
=> [2,5,3,4,6,8,7,1] => ? => ? = 1 - 1
{{1,8},{2,4,5},{3},{6,7}}
=> [8,4,3,5,2,7,6,1] => ? => ? = 1 - 1
{{1,8},{2,6,7},{3},{4,5}}
=> [8,6,3,5,4,7,2,1] => ? => ? = 1 - 1
{{1,2,8},{3},{4,5},{6,7}}
=> [2,8,3,5,4,7,6,1] => ? => ? = 1 - 1
{{1,2,6,8},{3},{4,5},{7}}
=> [2,6,3,5,4,8,7,1] => ? => ? = 1 - 1
{{1,2,4,6},{3},{5},{7,8}}
=> [2,4,3,6,5,1,8,7] => ? => ? = 2 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000383
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> [1] => ([],1)
=> [1] => 1
{{1,2}}
=> [2,1] => ([(0,1)],2)
=> [1,1] => 1
{{1},{2}}
=> [1,2] => ([],2)
=> [2] => 2
{{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1
{{1,2},{3}}
=> [2,1,3] => ([(1,2)],3)
=> [1,2] => 2
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1
{{1},{2,3}}
=> [1,3,2] => ([(1,2)],3)
=> [1,2] => 2
{{1},{2},{3}}
=> [1,2,3] => ([],3)
=> [3] => 3
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> [1,3] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> [1,3] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> [1,3] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> [4] => 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,3] => 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,3] => 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> [1,4] => 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => ([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,1,1,1,2] => ? = 2
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => ([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,1,1,1,2] => ? = 2
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => ? = 2
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,1,1,1,1] => ? = 1
{{1,2},{3,7},{4,5},{6,8}}
=> [2,1,7,5,4,8,3,6] => ([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,1,1,1,2] => ? = 2
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> [2,4,1,1] => ? = 1
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,1,1] => ? = 1
{{1,4},{2,6},{3,7},{5,8}}
=> [4,6,7,1,8,2,3,5] => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> [1,4,1,1,1] => ? = 1
{{1,4},{2,3},{5,7},{6,8}}
=> [4,3,2,1,7,8,5,6] => ([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2] => ? = 2
{{1,3},{2,4},{5,8},{6,7}}
=> [3,4,1,2,8,7,6,5] => ([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2] => ? = 2
{{1,4},{2,3},{5,8},{6,7}}
=> [4,3,2,1,8,7,6,5] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8)
=> [6,2] => ? = 2
{{1,5},{2,3},{4,8},{6,7}}
=> [5,3,2,8,1,7,6,4] => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> [1,1,4,1,1] => ? = 1
{{1,8},{2,4},{3,7},{5,6}}
=> [8,4,7,2,6,5,3,1] => ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,1,1,1,1] => ? = 1
{{1,2},{3,5},{4,8},{6,7}}
=> [2,1,5,8,3,7,6,4] => ([(0,1),(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,1,1,1,2] => ? = 2
{{1,2},{3,8},{4,7},{5,6}}
=> [2,1,8,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => ? = 2
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> [2,1,4,3,6,5,8,7,10,9] => ([(0,9),(1,8),(2,7),(3,6),(4,5)],10)
=> ? => ? = 5
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => ([],8)
=> [8] => ? = 8
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? => ? = 7
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? => ? = 6
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? => ? = 6
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [1,2,3,4,8,6,7,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 5
{{1},{2},{3},{4},{5,7,8},{6}}
=> [1,2,3,4,7,6,8,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 5
{{1},{2},{3},{4},{5,8},{6,7}}
=> [1,2,3,4,8,7,6,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,5] => ? = 5
{{1},{2},{3},{4},{5,6,8},{7}}
=> [1,2,3,4,6,8,7,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 5
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? => ? = 5
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [1,2,3,8,5,6,7,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 4
{{1},{2},{3},{4,6,7,8},{5}}
=> [1,2,3,6,5,7,8,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 4
{{1},{2},{3},{4,7,8},{5,6}}
=> [1,2,3,7,6,5,8,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 4
{{1},{2},{3},{4,8},{5,6,7}}
=> [1,2,3,8,6,7,5,4] => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 4
{{1},{2},{3},{4,5,6,7,8}}
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> [1,3,4] => ? = 4
{{1},{2},{3,4},{5,6,7,8}}
=> [1,2,4,3,6,7,8,5] => ([(2,3),(4,7),(5,7),(6,7)],8)
=> ? => ? = 4
{{1},{2},{3,8},{4},{5,6,7}}
=> [1,2,8,4,6,7,5,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [2,1,1,1,3] => ? = 3
{{1},{2},{3,7,8},{4,6},{5}}
=> [1,2,7,6,5,4,8,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2},{3,8},{4,6,7},{5}}
=> [1,2,8,6,5,7,4,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2},{3,8},{4,7},{5,6}}
=> [1,2,8,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2},{3,4,5,6,7,8}}
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2,3},{4,5},{6,7},{8}}
=> [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
=> ? => ? = 5
{{1},{2,3},{4,8},{5,6},{7}}
=> [1,3,2,8,6,5,7,4] => ([(1,2),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2,3},{4,8},{5,7},{6}}
=> [1,3,2,8,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2,3},{4,8},{5,6,7}}
=> [1,3,2,8,6,7,5,4] => ([(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2,4},{3},{5},{6,7,8}}
=> [1,4,3,2,5,7,8,6] => ([(2,7),(3,7),(4,5),(4,6),(5,6)],8)
=> ? => ? = 4
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [1,8,3,4,5,6,7,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
{{1},{2,4,5},{3},{6,7},{8}}
=> [1,4,3,5,2,7,6,8] => ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 4
{{1},{2,4,6,7},{3},{5},{8}}
=> [1,4,3,6,5,7,2,8] => ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
{{1},{2,4,5,6,7,8},{3}}
=> [1,4,3,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
{{1},{2,3,4},{5},{6,7,8}}
=> [1,3,4,2,5,7,8,6] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> [2,2,4] => ? = 4
{{1},{2,8},{3,4},{5,6},{7}}
=> [1,8,4,3,6,5,7,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
{{1},{2,8},{3,4},{5,6,7}}
=> [1,8,4,3,6,7,5,2] => ([(1,2),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
{{1},{2,5,6,7,8},{3,4}}
=> [1,5,4,3,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
{{1},{2,6},{3,5},{4},{7,8}}
=> [1,6,5,4,3,2,8,7] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
Description
The last part of an integer composition.
Matching statistic: St000010
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => ([],1)
=> [1]
=> 1
{{1,2}}
=> [2,1] => ([(0,1)],2)
=> [2]
=> 1
{{1},{2}}
=> [1,2] => ([],2)
=> [1,1]
=> 2
{{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
{{1,2},{3}}
=> [2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
{{1},{2,3}}
=> [1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2
{{1},{2},{3}}
=> [1,2,3] => ([],3)
=> [1,1,1]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> [2,1,4,3,6,5,8,7,10,9] => ([(0,9),(1,8),(2,7),(3,6),(4,5)],10)
=> ?
=> ? = 5
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ?
=> ? = 7
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ?
=> ? = 6
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [1,2,3,4,8,6,7,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
{{1},{2},{3},{4},{5,7,8},{6}}
=> [1,2,3,4,7,6,8,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
{{1},{2},{3},{4},{5,6,8},{7}}
=> [1,2,3,4,6,8,7,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 5
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [1,2,3,8,5,6,7,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
{{1},{2},{3},{4,6,7,8},{5}}
=> [1,2,3,6,5,7,8,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
{{1},{2},{3},{4,7,8},{5,6}}
=> [1,2,3,7,6,5,8,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
{{1},{2},{3},{4,8},{5,6,7}}
=> [1,2,3,8,6,7,5,4] => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
{{1},{2},{3,4},{5,6,7,8}}
=> [1,2,4,3,6,7,8,5] => ([(2,3),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 4
{{1},{2},{3,7,8},{4,6},{5}}
=> [1,2,7,6,5,4,8,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2},{3,8},{4,6,7},{5}}
=> [1,2,8,6,5,7,4,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2},{3,8},{4,7},{5,6}}
=> [1,2,8,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2},{3,4,5,6,7,8}}
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2,3},{4,5},{6,7},{8}}
=> [1,3,2,5,4,7,6,8] => ([(2,7),(3,6),(4,5)],8)
=> ?
=> ? = 5
{{1},{2,3},{4,8},{5,6},{7}}
=> [1,3,2,8,6,5,7,4] => ([(1,2),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2,3},{4,8},{5,7},{6}}
=> [1,3,2,8,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2,3},{4,8},{5,6,7}}
=> [1,3,2,8,6,7,5,4] => ([(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2,4},{3},{5},{6,7,8}}
=> [1,4,3,2,5,7,8,6] => ([(2,7),(3,7),(4,5),(4,6),(5,6)],8)
=> ?
=> ? = 4
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [1,8,3,4,5,6,7,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1},{2,4,5},{3},{6,7},{8}}
=> [1,4,3,5,2,7,6,8] => ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
{{1},{2,4,6,7},{3},{5},{8}}
=> [1,4,3,6,5,7,2,8] => ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2,4,5,6,7,8},{3}}
=> [1,4,3,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1},{2,8},{3,4},{5,6},{7}}
=> [1,8,4,3,6,5,7,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1},{2,8},{3,4},{5,6,7}}
=> [1,8,4,3,6,7,5,2] => ([(1,2),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1},{2,5,6,7,8},{3,4}}
=> [1,5,4,3,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1},{2,6},{3,5},{4},{7,8}}
=> [1,6,5,4,3,2,8,7] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2,8},{3,5},{4},{6,7}}
=> [1,8,5,4,3,7,6,2] => ([(1,2),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1},{2,6,7,8},{3,5},{4}}
=> [1,6,5,4,3,7,8,2] => ?
=> ?
=> ? = 2
{{1},{2,7},{3,6},{4,5},{8}}
=> [1,7,6,5,4,3,2,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
{{1},{2,3,4,5,6,8},{7}}
=> [1,3,4,5,6,8,7,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1},{2,3,4,5,6,7,8}}
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3},{4},{5},{6},{7},{8}}
=> [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ?
=> ? = 7
{{1,2},{3},{4},{5,6,7,8}}
=> [2,1,3,4,6,7,8,5] => ([(2,3),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 4
{{1,2},{3,5},{4},{6},{7},{8}}
=> [2,1,5,4,3,6,7,8] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
{{1,2},{3,7,8},{4},{5,6}}
=> [2,1,7,4,6,5,8,3] => ([(0,1),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,8},{4},{5,6,7}}
=> [2,1,8,4,6,7,5,3] => ([(0,1),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,5,6,7,8},{4}}
=> [2,1,5,4,6,7,8,3] => ([(0,1),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,4,5},{6},{7},{8}}
=> [2,1,4,5,3,6,7,8] => ([(3,4),(5,7),(6,7)],8)
=> ?
=> ? = 5
{{1,2},{3,7,8},{4,6},{5}}
=> [2,1,7,6,5,4,8,3] => ([(0,7),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,8},{4,6,7},{5}}
=> [2,1,8,6,5,7,4,3] => ([(0,1),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,4,8},{5},{6,7}}
=> [2,1,4,8,5,7,6,3] => ([(0,1),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,8},{4,5,6},{7}}
=> [2,1,8,5,6,4,7,3] => ([(0,1),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,4,8},{5,6},{7}}
=> [2,1,4,8,6,5,7,3] => ([(0,1),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,8},{4,5,7},{6}}
=> [2,1,8,5,7,6,4,3] => ([(0,1),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,4,8},{5,7},{6}}
=> [2,1,4,8,7,6,5,3] => ([(0,7),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
{{1,2},{3,4,5,8},{6},{7}}
=> [2,1,4,5,8,6,7,3] => ([(0,1),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
Description
The length of the partition.
Matching statistic: St000675
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> ? = 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 3
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 1
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 1
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 1
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 1
{{1,6},{2,7},{3,4},{5,8}}
=> [6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,7},{2,8},{3,5},{4,6}}
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,7},{4,6},{5,8}}
=> [3,7,1,6,8,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 1
{{1,6},{2,7},{3,5},{4,8}}
=> [6,7,5,8,3,1,2,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 1
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 1
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,7},{2,5},{3,6},{4,8}}
=> [7,5,6,8,2,3,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 1
{{1,6},{2,5},{3,7},{4,8}}
=> [6,5,7,8,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
{{1,5},{2,6},{3,7},{4,8}}
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,4},{2,6},{3,7},{5,8}}
=> [4,6,7,1,8,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 1
{{1,3},{2,6},{4,7},{5,8}}
=> [3,6,1,7,8,2,4,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 1
{{1,3},{2,5},{4,7},{6,8}}
=> [3,5,1,7,2,8,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,4},{2,5},{3,7},{6,8}}
=> [4,5,7,1,2,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 1
{{1,5},{2,4},{3,7},{6,8}}
=> [5,4,7,2,1,8,3,6] => [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> ? = 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
{{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 3
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
{{1,3},{2,5},{4,6},{7,8}}
=> [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
{{1,2},{3,5},{4,6},{7,8}}
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
{{1,2},{3,6},{4,5},{7,8}}
=> [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 3
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,2},{3,8},{4,5},{6,7}}
=> [2,1,8,5,4,7,6,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 2
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,2},{3,8},{4,6},{5,7}}
=> [2,1,8,6,7,4,5,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 2
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,7},{2,5},{3,6},{4,8}}
=> [7,5,6,8,2,3,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3},{2,5},{4,7},{6,8}}
=> [3,5,1,7,2,8,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,4},{2,5},{3,7},{6,8}}
=> [4,5,7,1,2,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,5},{2,4},{3,7},{6,8}}
=> [5,4,7,2,1,8,3,6] => [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,7},{2,4},{3,6},{5,8}}
=> [7,4,6,2,8,3,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,8},{2,4},{3,6},{5,7}}
=> [8,4,6,2,7,3,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000546
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => [1,2] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,3,1] => [1,3,2,4] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => 1 = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [1,3,4,2,5] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [2,1,3,5,4] => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 1 = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 2 = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [3,4,5,1,2] => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 3 = 4 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,4,2,3,5] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [5,1,2,3,4] => 1 = 2 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,4,3,1] => [1,3,4,2,5] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 2 = 3 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => ? = 1 - 1
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [6,2,3,4,5,1,7] => [7,1,5,4,3,2,6] => ? = 2 - 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [7,2,3,4,6,5,1] => [1,5,6,4,3,2,7] => ? = 1 - 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => [6,7,1,4,3,2,5] => ? = 2 - 1
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [5,2,3,4,1,6,7] => [7,6,1,4,3,2,5] => ? = 3 - 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [7,2,3,5,4,6,1] => [1,6,4,5,3,2,7] => ? = 1 - 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [6,2,3,7,5,1,4] => [4,1,5,7,3,2,6] => ? = 1 - 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [6,2,3,5,4,1,7] => [7,1,4,5,3,2,6] => ? = 2 - 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [4,2,3,1,7,6,5] => [5,6,7,1,3,2,4] => ? = 2 - 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [4,2,3,1,6,5,7] => [7,5,6,1,3,2,4] => ? = 3 - 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [4,2,3,1,7,6,5] => [5,6,7,1,3,2,4] => ? = 2 - 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => [6,7,5,1,3,2,4] => ? = 3 - 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [4,2,3,1,5,6,7] => [7,6,5,1,3,2,4] => ? = 4 - 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [7,2,4,3,5,6,1] => [1,6,5,3,4,2,7] => ? = 1 - 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [6,2,7,5,4,1,3] => [3,1,4,5,7,2,6] => ? = 1 - 1
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [6,2,4,3,5,1,7] => [7,1,5,3,4,2,6] => ? = 2 - 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => [3,6,1,4,7,2,5] => ? = 1 - 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [5,2,6,4,1,3,7] => [7,3,1,4,6,2,5] => ? = 2 - 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [7,2,4,3,6,5,1] => [1,5,6,3,4,2,7] => ? = 1 - 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [5,2,7,6,1,4,3] => [3,4,1,6,7,2,5] => ? = 1 - 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [5,2,4,3,1,7,6] => [6,7,1,3,4,2,5] => ? = 2 - 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [5,2,4,3,1,6,7] => [7,6,1,3,4,2,5] => ? = 3 - 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [7,2,5,4,3,6,1] => [1,6,3,4,5,2,7] => ? = 1 - 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [6,2,7,4,5,1,3] => [3,1,5,4,7,2,6] => ? = 1 - 1
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [6,2,5,4,3,1,7] => [7,1,3,4,5,2,6] => ? = 2 - 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [3,2,1,7,5,6,4] => [4,6,5,7,1,2,3] => ? = 2 - 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => ? = 2 - 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [7,2,5,4,3,6,1] => [1,6,3,4,5,2,7] => ? = 1 - 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [6,2,7,5,4,1,3] => [3,1,4,5,7,2,6] => ? = 1 - 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [6,2,7,4,5,1,3] => [3,1,5,4,7,2,6] => ? = 1 - 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [6,2,5,4,3,1,7] => [7,1,3,4,5,2,6] => ? = 2 - 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [3,2,1,7,5,6,4] => [4,6,5,7,1,2,3] => ? = 2 - 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => ? = 2 - 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => ? = 2 - 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [3,2,1,4,7,6,5] => [5,6,7,4,1,2,3] => ? = 3 - 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [3,2,1,4,6,5,7] => [7,5,6,4,1,2,3] => ? = 4 - 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => ? = 2 - 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [3,2,1,4,7,6,5] => [5,6,7,4,1,2,3] => ? = 3 - 1
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => [6,7,5,4,1,2,3] => ? = 4 - 1
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [7,3,2,4,5,6,1] => [1,6,5,4,2,3,7] => ? = 1 - 1
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => ? = 1 - 1
{{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => [6,3,2,4,5,1,7] => [7,1,5,4,2,3,6] => ? = 2 - 1
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [5,7,4,3,1,6,2] => [2,6,1,3,4,7,5] => ? = 1 - 1
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [5,6,4,3,1,2,7] => [7,2,1,3,4,6,5] => ? = 2 - 1
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [7,3,2,4,6,5,1] => [1,5,6,4,2,3,7] => ? = 1 - 1
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [5,7,6,4,1,3,2] => [2,3,1,4,6,7,5] => ? = 1 - 1
{{1,2,4,5},{3},{6,7}}
=> [2,4,3,5,1,7,6] => [5,3,2,4,1,7,6] => [6,7,1,4,2,3,5] => ? = 2 - 1
{{1,2,4,5},{3},{6},{7}}
=> [2,4,3,5,1,6,7] => [5,3,2,4,1,6,7] => [7,6,1,4,2,3,5] => ? = 3 - 1
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [7,5,3,4,2,6,1] => [1,6,2,4,3,5,7] => ? = 1 - 1
{{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => [6,7,3,4,5,1,2] => [2,1,5,4,3,7,6] => ? = 1 - 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000025
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 4
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
{{1,3},{2,5},{4,6},{7,8}}
=> [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 2
{{1,2},{3,5},{4,6},{7,8}}
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
{{1,2},{3,6},{4,5},{7,8}}
=> [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,6},{2,7},{3,4},{5,8}}
=> [6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 1
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,7},{4,5},{6,8}}
=> [2,1,7,5,4,8,3,6] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
{{1,2},{3,8},{4,5},{6,7}}
=> [2,1,8,5,4,7,6,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,7},{2,8},{3,5},{4,6}}
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,8},{4,6},{5,7}}
=> [2,1,8,6,7,4,5,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
{{1,2},{3,7},{4,6},{5,8}}
=> [2,1,7,6,8,4,3,5] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
{{1,3},{2,7},{4,6},{5,8}}
=> [3,7,1,6,8,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
{{1,6},{2,7},{3,5},{4,8}}
=> [6,7,5,8,3,1,2,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000007The number of saliances of the permutation. St000717The number of ordinal summands of a poset. St000553The number of blocks of a graph. St000843The decomposition number of a perfect matching. St001461The number of topologically connected components of the chord diagram of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000989The number of final rises of a permutation. St000056The decomposition (or block) number of a permutation. St000287The number of connected components of a graph. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000061The number of nodes on the left branch of a binary tree. St000181The number of connected components of the Hasse diagram for the poset. St001889The size of the connectivity set of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
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