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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000214
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Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000382
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 91%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> {{1}}
=> {{1}}
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> [1,1] => 1 = 0 + 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> [2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> [2,1] => 2 = 1 + 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> [1,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> [3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,2,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [2,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> [3,1,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> [3,1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> [2,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,5},{3}}
=> [2,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> [2,2,1] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> [4,1] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,5},{2},{3,4}}
=> [2,1,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> [2,1,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [3,2] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ?
=> ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ?
=> ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ?
=> ? => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> {{1},{2},{3,4,5,8},{6},{7}}
=> {{1,4,5},{2},{3,8},{6},{7}}
=> ? => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ?
=> ? => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ?
=> ? => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5,7},{6},{8}}
=> {{1},{2,4,7},{3},{5},{6},{8}}
=> ? => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> {{1,3,4,6,7},{2},{5},{8}}
=> ? => ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> {{1},{2,7,8},{3,4,5,6}}
=> ?
=> ? => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3},{5,8},{6},{7}}
=> {{1},{2},{3,4,8},{5},{6},{7}}
=> ? => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,4,7,8},{2},{3},{5},{6}}
=> ?
=> ? => ? = 1 + 1
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> {{1,2,3},{4,7,8},{5,6}}
=> ?
=> ? => ? = 4 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,4,5},{2,3},{6,7,8}}
=> ?
=> ? => ? = 4 + 1
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> {{1,7,8},{2,3},{4,5,6}}
=> ?
=> ? => ? = 4 + 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,5},{3},{4},{6},{7}}
=> {{1},{2},{3,5},{4},{6},{7,8}}
=> ? => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,3,4,5},{8}}
=> ?
=> ? => ? = 4 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,4,5,6}}
=> ?
=> ? => ? = 3 + 1
[]
=> {}
=> ?
=> ? => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,4},{3},{5},{6},{7},{8}}
=> {{1},{2,4},{3},{5},{6},{7},{8,9}}
=> ? => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> {{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1] => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2,3},{4},{5},{6},{7},{8},{9}}
=> {{1,3},{2},{4},{5},{6},{7},{8},{9,10}}
=> ? => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6},{7,8,9,10}}
=> ?
=> ? => ? = 7 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6,7,8},{9,10}}
=> ?
=> ? => ? = 7 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6},{7,8},{9,10}}
=> {{1,2,3,4,6,8,10},{5},{7},{9}}
=> ? => ? = 6 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6},{7,8,9,10}}
=> ?
=> ? => ? = 7 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6,7,8},{9,10}}
=> {{1,2,3,4,6,7,8,10},{5},{9}}
=> ? => ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8},{9}}
=> {{1},{2},{3},{4},{5},{6,7},{8},{9}}
=> ? => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,8},{2},{3},{4},{5},{6},{7},{9}}
=> {{1},{2},{3},{4},{5},{6},{7,8},{9}}
=> ? => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
=> {{1},{2},{3},{4},{5},{6},{7},{8,9},{10}}
=> ? => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
=> {{1,2,4,6,8,10,12},{3},{5},{7},{9},{11}}
=> [7,1,1,1,1,1] => ? = 6 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7,8,9,10},{11,12}}
=> ?
=> ? => ? = 10 + 1
Description
The first part of an integer composition.
Matching statistic: St001484
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 91%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,2]
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,2]
=> ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2]
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,2]
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2]
=> ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2]
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2]
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2]
=> ? = 4
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,1]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,1]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1]
=> ? = 5
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,1]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,1]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,1]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,1]
=> ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [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]
=> [7,6,5,4,3,1]
=> ? = 6
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1,1,1]
=> ? = 0
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,1,1]
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,1,1]
=> ? = 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,1,1]
=> ? = 3
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,2,1]
=> ? = 4
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> ? = 6
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2,1,1]
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,1,1]
=> ? = 2
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1,1]
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1,1]
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,4,2,1,1,1]
=> ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,1]
=> ? = 4
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [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]
=> [6,6,5,2,2,1]
=> ? = 2
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2,1]
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1]
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 6
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,2,1]
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> ? = 6
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [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]
=> [7,5,4,3,2,1]
=> ? = 6
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ? = 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ? = 3
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? = 3
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> ? = 5
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2]
=> ? = 7
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2]
=> ? = 5
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2]
=> ? = 8
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [7,5,5,4,3,2,1]
=> ? = 5
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,2]
=> ? = 5
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000441
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 91%
St000441: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 0
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => ? = 0
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,1,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => ? = 0
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => ? = 0
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => ? = 0
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? = 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,2,4,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,3,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,8,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,2,3,1] => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,2,3,8,1] => ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,2,3,8,1] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [6,7,2,3,4,5,8,1] => ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [6,7,5,8,3,4,1,2] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,4,1,2] => ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,6,1,2] => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,4,6,1,2] => ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,1,2] => ? = 3
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,6,8,1,2] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,2,1,6] => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,3,2,1,6] => ? = 1
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,1,8] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,2,3,1,4] => ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [6,7,5,8,2,3,1,4] => ? = 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,1,8] => ? = 3
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,1,2,3] => ? = 6
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,1,2,4] => ? = 3
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,1,2,4] => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,1,2,8] => ? = 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,1,2,5] => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,4,1,2,6] => ? = 3
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,1,2,8] => ? = 3
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,1,2,8] => ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,6,1,2,8] => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => ? = 4
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,2,1,3,8] => ? = 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,2,1,4,8] => ? = 0
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,1,7,8] => ? = 3
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,1,2,3,8] => ? = 2
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000932
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 77%●distinct values known / distinct values provided: 73%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 77%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,2,2,1]
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,2,1,1]
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ?
=> ? = 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,1,1]
=> ?
=> ? = 3
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,2,1]
=> ?
=> ? = 4
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2,1]
=> ?
=> ? = 3
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,1,1]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,1,1]
=> ?
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,2,2,1]
=> ?
=> ? = 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,1]
=> ?
=> ? = 3
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1,1]
=> ?
=> ? = 3
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,2,1]
=> ?
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,8,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
(load all 60 compositions to match this statistic)
(load all 60 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 70%●distinct values known / distinct values provided: 64%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 70%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ?
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,2,2,1]
=> ?
=> ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,2,1,1]
=> ?
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ?
=> ? = 5
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ?
=> ? = 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,1,1]
=> ?
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,2,1]
=> ?
=> ? = 4
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2,1]
=> ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,2,1,1]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,1,1]
=> ?
=> ? = 1
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000237
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,1,2] => [3,1,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,1,3,2] => [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,1,3] => [2,4,1,3] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,2,3] => [1,4,2,5,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,2,4,3] => [1,5,4,2,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,4,1,2,3] => [4,1,2,5,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,1,2,5] => [3,1,4,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,2,4,3] => [5,1,4,2,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,1,2,4] => [3,1,5,2,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,1,2] => [3,1,4,5,2] => 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [1,7,6,5,4,2,3] => [1,4,2,5,6,7,3] => ? = 3
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [1,7,5,3,2,4,6] => [1,3,5,2,7,4,6] => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => [1,7,6,5,3,2,4] => [1,3,5,2,6,7,4] => ? = 3
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [1,7,6,3,2,5,4] => [1,3,6,5,2,7,4] => ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [1,7,3,2,6,4,5] => [1,3,7,6,4,2,5] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [1,7,4,3,2,5,6] => [1,3,4,7,2,5,6] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [1,7,6,4,3,2,5] => [1,3,4,6,2,7,5] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [1,7,2,6,5,4,3] => [1,7,4,5,6,2,3] => ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [1,7,3,2,6,5,4] => [1,3,7,5,6,2,4] => ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [1,7,4,3,2,6,5] => [1,3,4,7,6,2,5] => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [1,7,5,4,3,2,6] => [1,3,4,5,7,2,6] => ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => ? = 5
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 5
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [6,5,4,3,1,2,7] => [3,1,4,5,6,2,7] => ? = 3
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => [7,1,5,2,3,4,6] => [7,5,2,3,1,4,6] => ? = 0
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => [7,1,6,2,3,4,5] => [7,6,2,3,4,1,5] => ? = 0
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => [6,4,2,1,3,5,7] => [2,4,1,6,3,5,7] => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [6,5,4,2,1,3,7] => [2,4,1,5,6,3,7] => ? = 3
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [4,3,5,2,6,7,1] => [7,1,5,2,4,3,6] => [7,5,4,2,1,3,6] => ? = 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => [6,5,2,1,4,3,7] => [2,5,4,1,6,3,7] => ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,3,6,2,1,7] => [6,2,1,5,3,4,7] => [2,6,5,3,1,4,7] => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => [6,3,2,1,4,5,7] => [2,3,6,1,4,5,7] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [6,5,3,2,1,4,7] => [2,3,5,1,6,4,7] => ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [6,1,5,4,3,2,7] => [6,3,4,5,1,2,7] => ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => [6,2,1,5,4,3,7] => [2,6,4,5,1,3,7] => ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [6,3,2,1,5,4,7] => [2,3,6,5,1,4,7] => ? = 3
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [6,4,3,2,1,5,7] => [2,3,4,6,1,5,7] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,5,4,3] => [1,2,8,5,4,3,6,7] => [1,2,4,5,8,3,6,7] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,8,7,2] => [1,8,7,2,4,3,6,5] => [1,7,4,2,6,3,8,5] => ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,3,8,7,6,5,2] => [1,8,7,6,5,2,4,3] => [1,5,4,2,6,7,8,3] => ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,7,6,4,3,2] => [1,8,7,6,4,3,2,5] => [1,3,4,6,2,7,8,5] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => [1,8,7,2,6,5,4,3] => [1,7,4,5,6,2,8,3] => ? = 4
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,5,4,3,2] => [1,8,5,4,3,2,6,7] => [1,3,4,5,8,2,6,7] => ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => [1,8,7,5,4,3,2,6] => [1,3,4,5,7,2,8,6] => ? = 4
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => [1,8,4,3,2,7,6,5] => [1,3,4,8,6,7,2,5] => ? = 4
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => [1,8,5,4,3,2,7,6] => [1,3,4,5,8,7,2,6] => ? = 4
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => [1,8,6,5,4,3,2,7] => [1,3,4,5,6,8,2,7] => ? = 4
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,3,4,5,6,7,8,2] => ? = 6
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,6,8,5] => [2,1,4,3,8,5,7,6] => [2,1,4,3,8,7,5,6] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => [2,1,4,3,6,7,8,5] => ? = 5
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,8,7] => [2,1,6,3,5,4,8,7] => [2,1,6,5,3,4,8,7] => ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => [2,1,4,5,6,3,8,7] => ? = 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,5,7,4,8,3] => [2,1,8,3,7,4,6,5] => [2,1,8,7,6,4,3,5] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => [2,1,4,5,6,7,8,3] => ? = 6
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,5,6,8,7,4,1] => [8,7,4,1,2,3,5,6] => [4,1,2,7,3,5,8,6] => ? = 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,3,1,7,6,8,5] => [4,3,1,2,8,5,7,6] => [3,1,4,2,8,7,5,6] => ? = 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,6,8,7,5,4,3,1] => [8,7,5,4,3,1,2,6] => [3,1,4,5,7,2,8,6] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,2,1,6,5,8,7,4] => [3,2,1,8,7,4,6,5] => [2,3,1,7,6,4,8,5] => ? = 4
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [2,3,1,5,6,7,8,4] => ? = 6
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000502
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 73%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1,0]
=> {{1}}
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,6},{4},{5},{7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 0
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,3},{4,5},{8}}
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,3,4,5},{8}}
=> ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4,5,6},{8}}
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,4,5,6,7},{2,3},{8}}
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5},{6},{8}}
=> ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5,6},{8}}
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,5,6,7},{4},{8}}
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,6,7},{5},{8}}
=> ? = 4
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,4},{2,3},{5,6},{7,8}}
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,3,4},{2},{5,6},{7,8}}
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 5
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> {{1,2},{3,6},{4,5},{7,8}}
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> {{1,2},{3,5,6},{4},{7,8}}
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> {{1,5,6},{2},{3,4},{7,8}}
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> {{1,3,5,6},{2},{4},{7,8}}
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> {{1,3},{2},{4},{5},{6,8},{7}}
=> ? = 0
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3},{5,8},{6},{7}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 0
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> {{1,8},{2,6},{3,4},{5},{7}}
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,4},{2,3},{5,8},{6,7}}
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> {{1,3,4},{2},{5,8},{6,7}}
=> ? = 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,7},{5,6}}
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 4
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000504
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 73%
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> {{1}}
=> {{1}}
=> {{1}}
=> ? = 0 + 1
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 1 = 0 + 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2},{4,5}}
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ?
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> {{1,4,6},{2},{3},{5},{7},{8}}
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> {{1},{2},{3,4,5,8},{6},{7}}
=> {{1,4,5,6},{2},{3},{7},{8}}
=> {{1,5,6},{2},{3,4},{7},{8}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> {{1,3,7},{2},{4},{5},{6},{8}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ?
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6,7},{8}}
=> {{1},{2,3},{4,5},{6,7},{8}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5,7},{6},{8}}
=> {{1},{2,4},{3},{5,7},{6},{8}}
=> {{1},{2,4,7},{3},{5},{6},{8}}
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,6,8},{3},{5},{7}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> {{1},{2,3,5,7},{4},{6},{8}}
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7},{8}}
=> {{1,3,4,6,7},{2},{5},{8}}
=> ? = 4 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> {{1},{2,7,8},{3,4},{5,6}}
=> {{1,2,7},{3,4},{5,6},{8}}
=> {{1,2,4,6},{3},{5,7},{8}}
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3,4}}
=> {{1,2,3,4,7},{5,6},{8}}
=> {{1,2,3,4,6},{5,7},{8}}
=> ? = 4 + 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8},{5}}
=> {{1,2,3,5,6,7},{4},{8}}
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> {{1},{2,7,8},{3,4,5,6}}
=> ?
=> ?
=> ? = 4 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6},{7}}
=> {{1,4,5,6,7},{2},{3},{8}}
=> {{1,5,6,7},{2},{3,4},{8}}
=> ? = 3 + 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> {{1,2,4,5,6,7},{3},{8}}
=> {{1,2,5,6,7},{3,4},{8}}
=> ? = 4 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,6,7}}
=> {{1,5,6,7},{2,3,4},{8}}
=> {{1,3,4,6,7},{2,5},{8}}
=> ? = 4 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> {{1,4,5,6,7},{2,3},{8}}
=> {{1,3,5,6,7},{2,4},{8}}
=> ? = 4 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,4,5,6,7},{2},{8}}
=> {{1,4,5,6,7},{2,3},{8}}
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 4 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> {{1,2},{3,4},{5,7,8},{6}}
=> {{1,2,4},{3},{5,6},{7,8}}
=> {{1,2,6,8},{3,4},{5},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3,4},{5,8},{6,7}}
=> {{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6},{7,8}}
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> {{1,2},{3,5,6},{4},{7,8}}
=> {{1,2},{3,4,6},{5},{7,8}}
=> {{1,2,4,8},{3,6},{5},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> {{1,2},{3,6},{4,5},{7,8}}
=> {{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> {{1,2},{3,8},{4,5},{6,7}}
=> {{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6},{7,8}}
=> {{1,2,4,5,6,8},{3},{7}}
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> {{1,2},{3,8},{4,7},{5,6}}
=> {{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> {{1,3},{2},{4},{5},{6,8},{7}}
=> {{1,3},{2},{4},{5},{6,8},{7}}
=> {{1},{2,3,8},{4},{5},{6},{7}}
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3},{5,8},{6},{7}}
=> {{1,4},{2},{3},{5,8},{6},{7}}
=> {{1},{2},{3,4,8},{5},{6},{7}}
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> {{1,6},{2},{3},{4},{5},{7},{8}}
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> {{1,6},{2},{3},{4},{5},{7,8}}
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> {{1,2},{3},{4},{5},{6,8},{7}}
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,4,7,8},{2},{3},{5},{6}}
=> ?
=> ?
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,3,4},{2},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6,8},{7}}
=> {{1,2,4,6},{3,8},{5},{7}}
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> {{1,3,4},{2},{5,7,8},{6}}
=> {{1,2,4},{3},{5,6,8},{7}}
=> {{1,2,6},{3,4,8},{5},{7}}
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> {{1,3,4},{2},{5,8},{6,7}}
=> {{1,4},{2,3},{5,6,8},{7}}
=> {{1,3,6},{2,4,8},{5},{7}}
=> ? = 2 + 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,3,4,5,7,8},{2},{6}}
=> {{1,2,4,5,6,8},{3},{7}}
=> {{1,2,5,6},{3,4,8},{7}}
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3,7,8},{4},{5},{6}}
=> ? = 4 + 1
Description
The cardinality of the first block of a set partition.
The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000445
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 73%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 5
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 6
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 3
Description
The number of rises of length 1 of a Dyck path.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001061The number of indices that are both descents and recoils of a permutation. St000247The number of singleton blocks of a set partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000248The number of anti-singletons of a set partition. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000931The number of occurrences of the pattern UUU in a Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St000731The number of double exceedences of a permutation. St000365The number of double ascents of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001948The number of augmented double ascents of a permutation. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St001960The number of descents of a permutation minus one if its first entry is not one.
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