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Matching statistic: St000234
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> {{1}}
=> {{1}}
=> [1] => 0
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> [2,1] => 0
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> [1,2] => 1
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 0
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => 1
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => 1
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 2
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 0
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 1
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 1
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 0
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 2
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 2
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 2
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 3
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => 0
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 0
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 1
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => 0
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 2
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 2
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 2
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 2
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 3
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 3
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 3
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 3
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 4
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => 0
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 0
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => 0
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => 1
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 1
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => 0
Description
The number of global ascents of a permutation.
The global ascents are the integers i such that
C(π)={i∈[n−1]∣∀1≤j≤i<k≤n:π(j)<π(k)}.
Equivalently, by the pigeonhole principle,
C(π)={i∈[n−1]∣∀1≤j≤i:π(j)≤i}.
For n>1 it can also be described as an occurrence of the mesh pattern
([1,2],{(0,2),(1,0),(1,1),(2,0),(2,1)})
or equivalently
([1,2],{(0,1),(0,2),(1,1),(1,2),(2,0)}),
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000056
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 86%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 86%
Values
[[1]]
=> {{1}}
=> {{1}}
=> [1] => 1 = 0 + 1
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> [2,1] => 1 = 0 + 1
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> [1,2] => 2 = 1 + 1
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 1 = 0 + 1
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => 2 = 1 + 1
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => 2 = 1 + 1
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 3 = 2 + 1
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 1 = 0 + 1
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 1 = 0 + 1
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2 = 1 + 1
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 2 = 1 + 1
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 2 = 1 + 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 1 = 0 + 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 3 = 2 + 1
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 3 = 2 + 1
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 3 = 2 + 1
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 4 = 3 + 1
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1 = 0 + 1
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => 1 = 0 + 1
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 1 = 0 + 1
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 2 = 1 + 1
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 2 = 1 + 1
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 2 = 1 + 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1 = 0 + 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => 1 = 0 + 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 2 = 1 + 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1 = 0 + 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,5},{2},{3,4}}
=> [5,2,4,3,1] => 1 = 0 + 1
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 2 = 1 + 1
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 3 = 2 + 1
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 2 = 1 + 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 3 = 2 + 1
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 3 = 2 + 1
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 3 = 2 + 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 3 = 2 + 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 2 = 1 + 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 3 = 2 + 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 5 = 4 + 1
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => 1 = 0 + 1
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 1 = 0 + 1
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => 1 = 0 + 1
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => 1 = 0 + 1
[[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 0 + 1
[[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> {{1,4,5,6,7},{2,3}}
=> [4,3,2,5,6,7,1] => ? = 0 + 1
[[1,2,4,5,6,7],[3]]
=> {{1,2,4,5,6,7},{3}}
=> {{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => ? = 0 + 1
[[1,2,3,5,6,7],[4]]
=> {{1,2,3,5,6,7},{4}}
=> {{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => ? = 0 + 1
[[1,2,3,4,6,7],[5]]
=> {{1,2,3,4,6,7},{5}}
=> {{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => ? = 0 + 1
[[1,2,3,4,5,7],[6]]
=> {{1,2,3,4,5,7},{6}}
=> {{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => ? = 1 + 1
[[1,2,3,4,5,6],[7]]
=> {{1,2,3,4,5,6},{7}}
=> {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1 + 1
[[1,3,5,6,7],[2,4]]
=> {{1,3,5,6,7},{2,4}}
=> {{1,6,7},{2,3,4,5}}
=> [6,3,4,5,2,7,1] => ? = 0 + 1
[[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> {{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => ? = 0 + 1
[[1,3,4,6,7],[2,5]]
=> {{1,3,4,6,7},{2,5}}
=> {{1,4,7},{2,3,5,6}}
=> [4,3,5,7,6,2,1] => ? = 0 + 1
[[1,2,4,6,7],[3,5]]
=> {{1,2,4,6,7},{3,5}}
=> {{1,2,7},{3,4,5,6}}
=> [2,7,4,5,6,3,1] => ? = 0 + 1
[[1,2,3,6,7],[4,5]]
=> {{1,2,3,6,7},{4,5}}
=> {{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => ? = 0 + 1
[[1,3,4,5,7],[2,6]]
=> {{1,3,4,5,7},{2,6}}
=> {{1,4,5},{2,3,6,7}}
=> [4,3,6,5,1,7,2] => ? = 0 + 1
[[1,2,3,5,7],[4,6]]
=> {{1,2,3,5,7},{4,6}}
=> {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 1 + 1
[[1,2,3,4,7],[5,6]]
=> {{1,2,3,4,7},{5,6}}
=> {{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => ? = 0 + 1
[[1,2,4,5,6],[3,7]]
=> {{1,2,4,5,6},{3,7}}
=> {{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => ? = 0 + 1
[[1,2,3,5,6],[4,7]]
=> {{1,2,3,5,6},{4,7}}
=> {{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => ? = 0 + 1
[[1,2,3,4,6],[5,7]]
=> {{1,2,3,4,6},{5,7}}
=> {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 1 + 1
[[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> {{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => ? = 0 + 1
[[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> {{1,5,6,7},{2},{3,4}}
=> [5,2,4,3,6,7,1] => ? = 0 + 1
[[1,3,5,6,7],[2],[4]]
=> {{1,3,5,6,7},{2},{4}}
=> {{1,6,7},{2,3,5},{4}}
=> [6,3,5,4,2,7,1] => ? = 0 + 1
[[1,2,5,6,7],[3],[4]]
=> {{1,2,5,6,7},{3},{4}}
=> {{1,2,6,7},{3},{4,5}}
=> [2,6,3,5,4,7,1] => ? = 0 + 1
[[1,3,4,6,7],[2],[5]]
=> {{1,3,4,6,7},{2},{5}}
=> {{1,4,7},{2,3,6},{5}}
=> [4,3,6,7,5,2,1] => ? = 0 + 1
[[1,2,4,6,7],[3],[5]]
=> {{1,2,4,6,7},{3},{5}}
=> {{1,2,7},{3,4,6},{5}}
=> [2,7,4,6,5,3,1] => ? = 0 + 1
[[1,2,3,6,7],[4],[5]]
=> {{1,2,3,6,7},{4},{5}}
=> {{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => ? = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> {{1,2,3,4,7},{5},{6}}
=> {{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => ? = 2 + 1
[[1,3,4,5,6],[2],[7]]
=> {{1,3,4,5,6},{2},{7}}
=> {{1,4,5,6},{2,3},{7}}
=> [4,3,2,5,6,1,7] => ? = 1 + 1
[[1,2,3,4,6],[5],[7]]
=> {{1,2,3,4,6},{5},{7}}
=> {{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => ? = 2 + 1
[[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> {{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => ? = 2 + 1
[[1,3,5,7],[2,4,6]]
=> {{1,3,5,7},{2,4,6}}
=> {{1},{2,3,4,5,6,7}}
=> [1,3,4,5,6,7,2] => ? = 1 + 1
[[1,2,5,7],[3,4,6]]
=> {{1,2,5,7},{3,4,6}}
=> {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 0 + 1
[[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> {{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => ? = 0 + 1
[[1,3,5,6],[2,4,7]]
=> {{1,3,5,6},{2,4,7}}
=> {{1,6},{2,3,4,5,7}}
=> [6,3,4,5,7,1,2] => ? = 0 + 1
[[1,2,5,6],[3,4,7]]
=> {{1,2,5,6},{3,4,7}}
=> {{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => ? = 0 + 1
[[1,3,4,6],[2,5,7]]
=> {{1,3,4,6},{2,5,7}}
=> {{1,4},{2,3,5,6,7}}
=> [4,3,5,1,6,7,2] => ? = 0 + 1
[[1,2,4,6],[3,5,7]]
=> {{1,2,4,6},{3,5,7}}
=> {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 1 + 1
[[1,2,3,6],[4,5,7]]
=> {{1,2,3,6},{4,5,7}}
=> {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 0 + 1
[[1,3,4,5],[2,6,7]]
=> {{1,3,4,5},{2,6,7}}
=> {{1,4,5,7},{2,3,6}}
=> [4,3,6,5,7,2,1] => ? = 0 + 1
[[1,2,4,5],[3,6,7]]
=> {{1,2,4,5},{3,6,7}}
=> {{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => ? = 0 + 1
[[1,2,3,5],[4,6,7]]
=> {{1,2,3,5},{4,6,7}}
=> {{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => ? = 0 + 1
[[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> {{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => ? = 0 + 1
[[1,4,6,7],[2,5],[3]]
=> {{1,4,6,7},{2,5},{3}}
=> {{1,7},{2,6},{3,4,5}}
=> [7,6,4,5,3,2,1] => ? = 0 + 1
[[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> {{1,7},{2,3},{4,5,6}}
=> [7,3,2,5,6,4,1] => ? = 0 + 1
[[1,3,6,7],[2,4],[5]]
=> {{1,3,6,7},{2,4},{5}}
=> {{1,7},{2,3,4},{5,6}}
=> [7,3,4,2,6,5,1] => ? = 0 + 1
[[1,2,6,7],[3,4],[5]]
=> {{1,2,6,7},{3,4},{5}}
=> {{1,2,4,7},{3},{5,6}}
=> [2,4,3,7,6,5,1] => ? = 0 + 1
[[1,3,5,7],[2,6],[4]]
=> {{1,3,5,7},{2,6},{4}}
=> {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => ? = 1 + 1
[[1,2,4,7],[3,6],[5]]
=> {{1,2,4,7},{3,6},{5}}
=> {{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => ? = 2 + 1
[[1,2,3,7],[4,6],[5]]
=> {{1,2,3,7},{4,6},{5}}
=> {{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => ? = 1 + 1
[[1,3,5,7],[2,4],[6]]
=> {{1,3,5,7},{2,4},{6}}
=> {{1},{2,3,4,5,7},{6}}
=> [1,3,4,5,7,6,2] => ? = 1 + 1
[[1,2,5,7],[3,4],[6]]
=> {{1,2,5,7},{3,4},{6}}
=> {{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => ? = 0 + 1
Description
The decomposition (or block) number of a permutation.
For π∈Sn, this is given by
#{1≤k≤n:{π1,…,πk}={1,…,k}}.
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
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