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Matching statistic: St000493
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000493: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000493: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> 1
[1,1,0,0]
=> {{1,2}}
=> 0
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 6
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 5
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 4
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 9
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 8
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
Description
The los statistic of a set partition.
Let S=B1,…,Bk be a set partition with ordered blocks Bi and with minBa<minBb for a<b.
According to [1, Definition 3], a '''los''' (left-opener-smaller) of S is given by a pair i>j such that j=minBb and i∈Ba for a>b.
This is also the dual major index of [2].
Matching statistic: St000008
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> [1,1] => [1,1] => 1
[1,1,0,0]
=> {{1,2}}
=> [2] => [2] => 0
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1] => [1,1,1] => 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [1,2] => [2,1] => 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1] => [1,2] => 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1] => [1,2] => 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3] => [3] => 0
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => 6
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => 5
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => 4
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [1,3] => [3,1] => 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => 3
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2] => [2,2] => 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => 3
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1] => [1,3] => 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1] => [1,3] => 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2] => [2,2] => 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1] => [1,3] => 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4] => [4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,1,1,1,1] => 10
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [2,1,1,1] => 9
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,2,1,1] => 8
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,2,1,1] => 8
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [1,1,3] => [3,1,1] => 7
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,1,2,1] => 7
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => 6
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,1,2,1] => 7
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => [1,1,2,1] => 7
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [1,3,1] => [1,3,1] => 5
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [1,3,1] => [1,3,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [1,2,2] => [2,2,1] => 6
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [1,3,1] => [1,3,1] => 5
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [1,4] => [4,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => 6
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => 5
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => 4
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [2,3] => [3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => 6
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => 5
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,1,2] => 6
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => [1,1,1,2] => 6
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,1,2] => [2,1,2] => 5
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1] => [1,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => 3
Description
The major index of the composition.
The descents of a composition [c1,c2,…,ck] are the partial sums c1,c1+c2,…,c1+⋯+ck−1, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000009
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [[1,3,4],[2]]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,1,2] => [[1,4],[2],[3]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,1,3] => [[1,4],[2],[3]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[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]]
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 8
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,2,4,3] => [[1,3,4],[2],[5]]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => ? => ?
=> ? = 13
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,8,7,6,5,4,2] => ? => ?
=> ? = 8
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,7,6,5,4,3,2,8] => ? => ?
=> ? = 8
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,8,7,6,5,4,1] => ? => ?
=> ? = 3
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,8,7,6,5,4,3,1] => ? => ?
=> ? = 1
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,2,8,7,6,5,4,1] => ? => ?
=> ? = 2
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,8,7,6,5,4,2,1] => ? => ?
=> ? = 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [4,3,8,7,6,5,2,1] => ? => ?
=> ? = 2
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [5,4,8,7,6,3,2,1] => ? => ?
=> ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,7,5,4,3,2,1] => ? => ?
=> ? = 1
Description
The charge of a standard tableau.
Matching statistic: St000499
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000499: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
St000499: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 6
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 5
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 4
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 3
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 3
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[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}}
=> 10
[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}}
=> 9
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 8
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 8
[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}}
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 6
[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}}
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 5
[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}}
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> 5
[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
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> 6
[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}}
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,4},{3,5}}
=> {{1},{2,5},{3,4}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> {{1,5},{2},{3,4}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 3
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 13
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8},{3}}
=> {{1,3,5,7},{2,4,6},{8}}
=> {{1,7},{2,3,4,5,6},{8}}
=> ? = 8
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,5,6,7,8},{4}}
=> {{1,3,5},{2,4,6,7},{8}}
=> {{1,6,7},{2,3,4,5},{8}}
=> ? = 8
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8},{5}}
=> {{1,3,5,6,7},{2,4},{8}}
=> {{1,5,6,7},{2,3,4},{8}}
=> ? = 8
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> {{1,3},{2,4,5,6,7},{8}}
=> {{1,4,5,6,7},{2,3},{8}}
=> ? = 8
[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}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 8
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,4,5,6,7,8},{2},{3}}
=> {{1,4,7},{2,5,8},{3,6}}
=> {{1,8},{2,7},{3,4,5,6}}
=> ? = 3
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1,3,5,6,7,8},{2},{4}}
=> {{1,4,7},{2,5},{3,6,8}}
=> {{1,8},{2,6,7},{3,4,5}}
=> ? = 3
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,4,5,6,7,8},{3}}
=> {{1,3,5,7,8},{2,4,6}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,3,5},{2,4,6,7,8}}
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,6,7,8},{5}}
=> {{1,3,5,6,7,8},{2,4}}
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 1
Description
The rcb statistic of a set partition.
Let S=B1,…,Bk be a set partition with ordered blocks Bi and with minBa<minBb for a<b.
According to [1, Definition 3], a '''rcb''' (right-closer-bigger) of S is given by a pair i<j such that j=maxBb and i∈Ba for a<b.
Matching statistic: St000490
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000490: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000490: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 0
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 6
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 5
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 4
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,3},{2,4}}
=> 3
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 3
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 3
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 3
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,3,4},{2}}
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 9
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 8
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4},{2,5},{3}}
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,3},{2,5},{4}}
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1},{2,4},{3,5}}
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1,3},{2,4},{5}}
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2},{3,5}}
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,5},{2},{4}}
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,3,5},{2,4}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,4},{3,5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,4,5},{2},{3}}
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8}}
=> {{1,4,7},{2,5,8},{3,6}}
=> ? = 13
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8},{3}}
=> {{1},{2,4,6,8},{3,5,7}}
=> ? = 8
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,5,6,7,8},{4}}
=> {{1,3,5,7},{2},{4,6,8}}
=> ? = 8
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8},{5}}
=> {{1,3},{2,4,6,8},{5,7}}
=> ? = 8
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> {{1,3,5,7},{2,4},{6,8}}
=> ? = 8
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1},{2,3,4,5,6,7},{8}}
=> {{1,3,5,7},{2,4,6},{8}}
=> ? = 8
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,5},{2,4,6,8},{7}}
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 7
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8}}
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 6
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1,3,5,6,7,8},{2},{4}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 3
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,6,7,8},{5}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,7,8},{6}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,6,7}}
=> {{1,3,5,7,8},{2,4,6}}
=> ? = 6
Description
The intertwining number of a set partition.
This is defined in [1] as follows: for int(a,b)={min, the '''block intertwiners''' of two disjoint sets A,B of integers is given by
\{ (a,b) \in A\times B : \operatorname{int}(a,b) \cap A \cup B = \emptyset \}.
The intertwining number of a set partition S is now the number of intertwiners of all pairs of blocks of S.
Matching statistic: St000161
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 73%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 73%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [[.,.],[.,.]]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 3
[1,1,0,0,1,1,0,0]
=> [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
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 8
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 21
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? = 20
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 19
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 19
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 18
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 18
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 17
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 18
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 18
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ? = 16
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ? = 16
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 17
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ? = 16
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 17
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 16
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 15
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ? = 14
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 17
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 16
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 17
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 17
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,2,7,6,3,4,5] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 14
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,2,6,5,3,4,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 14
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,2,7,3,4,6,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 16
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,2,7,5,3,4,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 14
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,2,7,6,5,3,4] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ? = 12
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 14
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ? = 13
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,2,6,3,5,4,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,7,3,5,4,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 15
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,2,7,6,3,5,4] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,2,6,4,3,5,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 14
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,2,7,3,6,4,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 15
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,2,7,4,3,5,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 14
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,2,7,6,4,3,5] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ? = 12
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ? = 12
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,2,7,3,6,5,4] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ? = 14
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,2,7,4,3,6,5] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ? = 13
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,2,7,5,4,3,6] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ? = 12
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
=> ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 16
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ? = 15
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 14
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 14
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [.,[[.,.],[.,[[[.,.],.],.]]]]
=> ? = 13
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ? = 13
Description
The sum of the sizes of the right subtrees of a binary tree.
This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the 312-avoiding permutation corresponding to the binary tree.
It is also the sum of all heights j of the coordinates (i,j) of the Dyck path corresponding to the binary tree.
Matching statistic: St000833
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 73%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,1,2] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,1,3,2] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,1,3] => [1,3,4,2] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 8
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,2,3] => [5,1,2,4,3] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,2,4,3] => [5,1,4,2,3] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,2,4] => [5,1,3,4,2] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [4,5,1,2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,1,2,5] => [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,2,4,3] => [1,5,4,2,3] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,1,2,4] => [1,3,5,4,2] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,1,2] => [1,2,3,5,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 21
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 20
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 19
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 19
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 17
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 18
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 18
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 17
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 16
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 15
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 17
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 16
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 15
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 15
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 14
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 17
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 16
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 17
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 17
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,2,7,6,3,4,5] => [7,6,1,2,5,4,3] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,2,6,5,3,4,7] => [7,6,2,3,5,4,1] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => ? = 16
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,2,7,5,3,4,6] => [7,6,1,3,5,4,2] => ? = 14
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,2,7,6,5,3,4] => [7,6,1,2,3,5,4] => ? = 12
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 14
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 13
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,2,6,3,5,4,7] => [7,6,2,5,3,4,1] => ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => ? = 15
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,2,7,6,3,5,4] => [7,6,1,2,5,3,4] => ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,2,6,4,3,5,7] => [7,6,2,4,5,3,1] => ? = 14
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => ? = 15
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => ? = 14
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,2,7,6,4,3,5] => [7,6,1,2,4,5,3] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 14
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => ? = 13
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => ? = 12
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 16
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [7,5,6,4,3,1,2] => ? = 15
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 14
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => ? = 14
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [7,5,6,4,1,2,3] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => [7,5,6,2,4,3,1] => ? = 13
Description
The comajor index of a permutation.
This is, \operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i) for a permutation \pi of length n.
Matching statistic: St000018
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 77%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 77%
Values
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [2,4,3,1] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,2,4,1] => [3,2,4,1] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [1,4,3,2] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [1,3,4,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,2,1,3] => [2,4,1,3] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [3,5,4,2,1] => 8
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,3,5,2,1] => [4,3,5,2,1] => 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,5,2,1] => 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [2,5,4,3,1] => 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [2,4,5,3,1] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [3,5,2,4,1] => 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,3,2,5,1] => [4,3,2,5,1] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,2,4,5,1] => [3,2,4,5,1] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [2,3,5,4,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,4,2,5,1] => [3,4,2,5,1] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,2,3,5,1] => [2,4,3,5,1] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,5,4,3,2] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,4,5,3,2] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,3,5,4,2] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,3,5,1,2] => [1,4,3,5,2] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,3,4,5,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,2,1,3] => [2,5,4,1,3] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,2,1,3] => [2,4,5,1,3] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,3,2,1,4] => [3,5,2,1,4] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,2,1,4,5] => [3,2,1,4,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,2,1,3,4] => [2,1,5,3,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,4,2,1,5] => [3,4,2,1,5] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,1,3,5] => [2,4,1,3,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 20
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 19
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 19
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [4,6,7,5,3,2,1] => ? = 17
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,5,4,6,3,2,1] => [5,7,4,6,3,2,1] => ? = 18
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 18
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [5,4,6,7,3,2,1] => [5,4,6,7,3,2,1] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [4,5,7,6,3,2,1] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [5,6,4,7,3,2,1] => [5,6,4,7,3,2,1] => ? = 17
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [6,4,5,7,3,2,1] => [4,6,5,7,3,2,1] => ? = 16
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [4,5,6,7,3,2,1] => ? = 15
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => ? = 17
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [3,6,7,5,4,2,1] => ? = 16
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [3,5,7,6,4,2,1] => ? = 15
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [6,5,7,3,4,2,1] => [3,6,5,7,4,2,1] => ? = 15
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,5,6,7,4,2,1] => ? = 14
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [4,7,6,3,5,2,1] => ? = 17
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [6,7,4,3,5,2,1] => [4,6,7,3,5,2,1] => ? = 16
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,5,4,3,6,2,1] => [5,7,4,3,6,2,1] => ? = 17
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => ? = 17
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [5,4,3,6,7,2,1] => [5,4,3,6,7,2,1] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [7,4,3,5,6,2,1] => [4,3,7,5,6,2,1] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,6,4,3,7,2,1] => [5,6,4,3,7,2,1] => ? = 16
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [6,4,3,5,7,2,1] => [4,6,3,5,7,2,1] => ? = 14
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [4,3,5,6,7,2,1] => [4,3,5,6,7,2,1] => ? = 12
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => ? = 14
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,4,6,7,5,2,1] => ? = 13
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,4,5,3,6,2,1] => [4,5,7,3,6,2,1] => ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [6,4,5,3,7,2,1] => [4,6,5,3,7,2,1] => ? = 15
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [4,5,3,6,7,2,1] => [4,5,3,6,7,2,1] => ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,5,3,4,6,2,1] => [3,5,7,4,6,2,1] => ? = 14
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [5,4,6,3,7,2,1] => [5,4,6,3,7,2,1] => ? = 15
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [6,5,3,4,7,2,1] => [3,6,5,4,7,2,1] => ? = 14
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [5,3,4,6,7,2,1] => [3,5,4,6,7,2,1] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [4,5,6,3,7,2,1] => [4,5,6,3,7,2,1] => ? = 14
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [5,6,3,4,7,2,1] => [3,5,6,4,7,2,1] => ? = 13
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [6,3,4,5,7,2,1] => [3,4,6,5,7,2,1] => ? = 12
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [2,7,6,5,4,3,1] => ? = 16
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => [2,6,7,5,4,3,1] => ? = 15
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [2,5,7,6,4,3,1] => ? = 14
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [6,5,7,4,2,3,1] => [2,6,5,7,4,3,1] => ? = 14
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => [2,5,6,7,4,3,1] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [2,4,7,6,5,3,1] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [2,4,6,7,5,3,1] => ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,5,4,6,2,3,1] => [2,5,7,4,6,3,1] => ? = 13
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [6,5,4,7,2,3,1] => [2,6,5,4,7,3,1] => ? = 13
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions (i,i+1) needed to write \pi. Thus, it is also the Coxeter length of \pi.
Matching statistic: St000796
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000796: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 73%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000796: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 73%
Values
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,2,4,1] => [4,2,1,3] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,2,1,3] => [3,2,4,1] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => 8
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,3,5,2,1] => [5,4,2,1,3] => 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,3,2,5,1] => [5,3,2,1,4] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,2,4,5,1] => [5,2,1,3,4] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,4,2,5,1] => [5,3,1,2,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,2,3,5,1] => [5,2,3,1,4] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [4,5,2,3,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,3,5,1,2] => [4,5,2,1,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [4,5,1,2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,2,1,3] => [4,3,5,1,2] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,3,2,1,4] => [4,3,2,5,1] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,2,1,4,5] => [3,2,1,4,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,2,1,3,4] => [3,2,4,5,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,4,2,1,5] => [4,3,1,2,5] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,1,3,5] => [3,2,4,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 21
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 20
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,6,5,4,2,3,1] => ? = 19
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 19
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,5,3,4,2,1] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [7,6,5,3,4,1,2] => ? = 17
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,5,4,6,3,2,1] => [7,6,5,3,2,4,1] => ? = 18
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => ? = 18
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,6,5,2,3,4,1] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => ? = 17
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [6,4,5,7,3,2,1] => [7,6,5,2,3,1,4] => ? = 16
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 15
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,4,5,3,2,1] => ? = 17
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [7,6,4,5,3,1,2] => ? = 16
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,6,4,5,2,3,1] => ? = 15
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [6,5,7,3,4,2,1] => [7,6,4,5,2,1,3] => ? = 15
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [7,6,4,5,1,2,3] => ? = 14
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 17
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [6,7,4,3,5,2,1] => [7,6,4,3,5,1,2] => ? = 16
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,5,4,3,6,2,1] => [7,6,4,3,2,5,1] => ? = 17
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [6,5,4,3,7,2,1] => [7,6,4,3,2,1,5] => ? = 17
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [5,4,3,6,7,2,1] => [7,6,3,2,1,4,5] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [7,4,3,5,6,2,1] => [7,6,3,2,4,5,1] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,6,4,3,7,2,1] => [7,6,4,3,1,2,5] => ? = 16
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [6,4,3,5,7,2,1] => [7,6,3,2,4,1,5] => ? = 14
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [4,3,5,6,7,2,1] => [7,6,2,1,3,4,5] => ? = 12
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,3,4,5,2,1] => ? = 14
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [7,6,3,4,5,1,2] => ? = 13
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,4,5,3,6,2,1] => [7,6,4,2,3,5,1] => ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [6,4,5,3,7,2,1] => [7,6,4,2,3,1,5] => ? = 15
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [4,5,3,6,7,2,1] => [7,6,3,1,2,4,5] => ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,5,3,4,6,2,1] => [7,6,3,4,2,5,1] => ? = 14
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 15
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [6,5,3,4,7,2,1] => [7,6,3,4,2,1,5] => ? = 14
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [5,3,4,6,7,2,1] => [7,6,2,3,1,4,5] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [7,6,2,3,4,5,1] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 14
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 13
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [6,3,4,5,7,2,1] => [7,6,2,3,4,1,5] => ? = 12
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [7,5,6,4,3,2,1] => ? = 16
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => [7,5,6,4,3,1,2] => ? = 15
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => ? = 14
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [6,5,7,4,2,3,1] => [7,5,6,4,2,1,3] => ? = 14
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => [7,5,6,4,1,2,3] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [7,5,6,3,4,2,1] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [7,5,6,3,4,1,2] => ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,5,4,6,2,3,1] => [7,5,6,3,2,4,1] => ? = 13
Description
The stat' of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns (\underline{13}2), (\underline{31}2), (\underline{32}2) and (\underline{21}), where matches of the underlined letters must be adjacent.
Matching statistic: St000798
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000798: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 73%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000798: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 73%
Values
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 5
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,2,4,1] => [4,2,1,3] => 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,2,1,3] => [3,2,4,1] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => 8
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,3,5,2,1] => [5,4,2,1,3] => 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => 7
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,3,2,5,1] => [5,3,2,1,4] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,2,4,5,1] => [5,2,1,3,4] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,4,2,5,1] => [5,3,1,2,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,2,3,5,1] => [5,2,3,1,4] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 5
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [4,5,2,3,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,3,5,1,2] => [4,5,2,1,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [4,5,1,2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,2,1,3] => [4,3,5,1,2] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,3,2,1,4] => [4,3,2,5,1] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,2,1,4,5] => [3,2,1,4,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,2,1,3,4] => [3,2,4,5,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,4,2,1,5] => [4,3,1,2,5] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,1,3,5] => [3,2,4,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 21
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 20
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,6,5,4,2,3,1] => ? = 19
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 19
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,5,3,4,2,1] => ? = 18
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [7,6,5,3,4,1,2] => ? = 17
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,5,4,6,3,2,1] => [7,6,5,3,2,4,1] => ? = 18
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => ? = 18
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,6,5,2,3,4,1] => ? = 16
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => ? = 17
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [6,4,5,7,3,2,1] => [7,6,5,2,3,1,4] => ? = 16
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 15
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,4,5,3,2,1] => ? = 17
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [7,6,4,5,3,1,2] => ? = 16
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,6,4,5,2,3,1] => ? = 15
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [6,5,7,3,4,2,1] => [7,6,4,5,2,1,3] => ? = 15
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [7,6,4,5,1,2,3] => ? = 14
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 17
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [6,7,4,3,5,2,1] => [7,6,4,3,5,1,2] => ? = 16
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,5,4,3,6,2,1] => [7,6,4,3,2,5,1] => ? = 17
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [6,5,4,3,7,2,1] => [7,6,4,3,2,1,5] => ? = 17
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [5,4,3,6,7,2,1] => [7,6,3,2,1,4,5] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [7,4,3,5,6,2,1] => [7,6,3,2,4,5,1] => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,6,4,3,7,2,1] => [7,6,4,3,1,2,5] => ? = 16
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [6,4,3,5,7,2,1] => [7,6,3,2,4,1,5] => ? = 14
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [4,3,5,6,7,2,1] => [7,6,2,1,3,4,5] => ? = 12
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,3,4,5,2,1] => ? = 14
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [7,6,3,4,5,1,2] => ? = 13
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,4,5,3,6,2,1] => [7,6,4,2,3,5,1] => ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [6,4,5,3,7,2,1] => [7,6,4,2,3,1,5] => ? = 15
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [4,5,3,6,7,2,1] => [7,6,3,1,2,4,5] => ? = 13
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,5,3,4,6,2,1] => [7,6,3,4,2,5,1] => ? = 14
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 15
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [6,5,3,4,7,2,1] => [7,6,3,4,2,1,5] => ? = 14
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [5,3,4,6,7,2,1] => [7,6,2,3,1,4,5] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [7,6,2,3,4,5,1] => ? = 12
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 14
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 13
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [6,3,4,5,7,2,1] => [7,6,2,3,4,1,5] => ? = 12
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [7,5,6,4,3,2,1] => ? = 16
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => [7,5,6,4,3,1,2] => ? = 15
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => ? = 14
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [6,5,7,4,2,3,1] => [7,5,6,4,2,1,3] => ? = 14
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => [7,5,6,4,1,2,3] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [7,5,6,3,4,2,1] => ? = 13
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [7,5,6,3,4,1,2] => ? = 12
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,5,4,6,2,3,1] => [7,5,6,3,2,4,1] => ? = 13
Description
The makl of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns (1\underline{32}), (\underline{31}2), (\underline{32}1) and (\underline{21}), where matches of the underlined letters must be adjacent.
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