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Your data matches 38 different statistics following compositions of up to 3 maps.
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Matching statistic: St000251
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,1,2] => {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 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]
=> [4,3,2,1] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => {{1,2,4},{3}}
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,1,2] => {{1,2,3,4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 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]
=> [5,4,3,2,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,4,3] => {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,3,2] => {{1,2,5},{3,4}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,1,3,4,2] => {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,1,2,4] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,1,4,2] => {{1,2,3,5},{4}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => {{1,5},{2,3,4}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,5,1,4,3] => {{1,2,3,5},{4}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => {{1,2,3,4,5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,5,2,4] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,5,4,2] => {{1,2,3,5},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,3,1,5,2] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => {{1,2,3,4},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [2,3,5,1,4] => {{1,2,3,4,5}}
=> 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000254
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,1,2] => {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 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]
=> [4,3,2,1] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => {{1,2,4},{3}}
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,1,2] => {{1,2,3,4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 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]
=> [5,4,3,2,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,4,3] => {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,3,2] => {{1,2,5},{3,4}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,1,3,4,2] => {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,1,2,4] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,1,4,2] => {{1,2,3,5},{4}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => {{1,5},{2,3,4}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,5,1,4,3] => {{1,2,3,5},{4}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => {{1,2,3,4,5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,5,2,4] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,5,4,2] => {{1,2,3,5},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,3,1,5,2] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => {{1,2,3,4},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [2,3,5,1,4] => {{1,2,3,4,5}}
=> 1
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St001280
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,1,2] => [3]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [3]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,1,2,3] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,1,2] => [4]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,4,1,3] => [4]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [3,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => [4]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1,2,4] => [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [4]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,4,3] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,3,2] => [3,2]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,1,3,4,2] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,1,2,4] => [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,1,4,2] => [4,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => [3,2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => [5]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => [3,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,5,1,3,4] => [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,5,1,4,3] => [4,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,5,4,3,1] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [3,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,5,2,4] => [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,5,4,2] => [4,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,5,3] => [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,3,1,5,2] => [5]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => [4,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [2,3,5,1,4] => [5]
=> 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => [8,1,4,7,6,3,2,5] => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,6,2,1] => [8,1,4,5,6,2,7,3] => ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,2,7,1] => [8,7,4,5,2,3,1,6] => ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => [8,7,4,6,3,2,1,5] => ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,2,5,6,1] => [8,5,2,3,6,1,4,7] => ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,4,2,5,8,1] => [8,5,4,2,7,3,6,1] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,2,6,7,1] => [8,6,5,3,2,7,1,4] => ?
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [7,5,6,2,3,4,8,1] => [8,3,4,7,6,2,5,1] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [8,6,4,2,3,5,7,1] => [8,3,5,2,7,4,1,6] => ?
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,3,5,1,2] => [2,8,5,3,1,7,6,4] => ?
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,3,7,1,2] => [2,8,7,6,4,3,1,5] => ?
=> ? = 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,4,5,1,2] => [2,8,4,5,1,7,6,3] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,2,1,6] => [6,1,4,5,2,8,7,3] => ?
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,6,2,1,7] => [7,1,5,3,6,2,8,4] => ?
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,2,5,1,6] => [6,5,2,3,1,8,7,4] => ?
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [7,5,2,3,4,6,1,8] => [7,3,4,6,2,1,5,8] => ?
=> ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,6,1,2,3] => [2,3,8,6,4,1,5,7] => ?
=> ? = 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,1,2,3] => [2,3,8,7,4,5,1,6] => ?
=> ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,3,1,2,4] => [2,4,1,8,3,5,7,6] => ?
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,5,1,2,6] => [2,6,5,3,1,8,7,4] => ?
=> ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,1,2,6] => [2,6,4,5,1,8,7,3] => ?
=> ? = 1
[1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,2,1,4,5] => [4,1,2,5,8,7,6,3] => ?
=> ? = 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,1,2,3,5] => [2,3,5,1,8,7,6,4] => ?
=> ? = 2
[1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [7,6,4,3,1,2,5,8] => [2,5,1,3,7,4,6,8] => ?
=> ? = 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,1,2,6,8] => [2,6,4,1,3,7,5,8] => ?
=> ? = 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [7,8,2,3,1,4,5,6] => [4,3,1,5,6,8,7,2] => ?
=> ? = 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [7,3,4,1,2,5,6,8] => [2,5,4,1,6,7,3,8] => ?
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [7,5,1,2,3,4,6,8] => [2,3,4,6,1,7,5,8] => ?
=> ? = 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000035
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,3,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,5,6,7,2,1] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000659
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,3,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,3,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [8,7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [8,6,7,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,5,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,5,6,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,6,4,3,5,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,5,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,6,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,7,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,3,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000340
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,4,3,2,1] => [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]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,3,2,1] => [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
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,3,2,1] => [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
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,3,2,1] => [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]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,3,4,2,1] => [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
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,3,4,2,1] => [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]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,4,2,1] => [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
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,3,5,2,1] => [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
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,3,6,2,1] => [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
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,3,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,3,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [8,7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,2,1] => [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
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [8,6,7,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,5,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,5,6,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,6,4,3,5,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,5,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,6,2,1] => [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
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,2,3,1] => [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
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,4,2,3,1] => [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]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000245
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => [2,1,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => [2,1,5,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => [3,2,5,1,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => [2,5,4,1,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => [3,5,2,1,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,5,4,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => [3,2,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [4,3,2,5,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => [3,2,7,1,6,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => [3,7,2,1,6,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [3,2,4,5,1,6,7] => [3,2,7,6,1,5,4] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [2,4,3,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => [3,7,2,6,1,5,4] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [4,3,2,5,1,6,7] => [4,3,2,7,1,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => [3,2,7,6,1,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [3,4,5,2,1,6,7] => [3,7,6,2,1,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [4,3,5,2,1,6,7] => [4,3,7,2,1,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [2,5,4,3,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [3,5,4,2,1,6,7] => [3,7,6,2,1,5,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => [4,7,3,2,1,6,5] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [3,2,4,1,6,5,7] => [3,2,7,1,6,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [2,4,3,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [3,4,2,1,6,5,7] => [3,7,2,1,6,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [3,2,4,5,6,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [3,4,2,5,6,1,7] => [3,7,2,6,5,1,4] => ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [3,2,5,4,6,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [3,4,5,2,6,1,7] => [3,7,6,2,5,1,4] => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [4,3,5,2,6,1,7] => [4,3,7,2,6,1,5] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [2,5,4,3,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [3,5,4,2,6,1,7] => [3,7,6,2,5,1,4] => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [4,5,3,2,6,1,7] => [4,7,3,2,6,1,5] => ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => [5,4,3,2,7,1,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => [2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [3,2,4,6,5,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => ? = 2
Description
The number of ascents of a permutation.
Matching statistic: St000672
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => [2,1,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => [2,1,5,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => [3,2,5,1,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => [2,5,4,1,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => [3,5,2,1,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,5,4,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => [3,2,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [4,3,2,5,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => [3,2,7,1,6,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => [3,7,2,1,6,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [3,2,4,5,1,6,7] => [3,2,7,6,1,5,4] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [2,4,3,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => [3,7,2,6,1,5,4] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [4,3,2,5,1,6,7] => [4,3,2,7,1,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => [3,2,7,6,1,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [3,4,5,2,1,6,7] => [3,7,6,2,1,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [4,3,5,2,1,6,7] => [4,3,7,2,1,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [2,5,4,3,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [3,5,4,2,1,6,7] => [3,7,6,2,1,5,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => [4,7,3,2,1,6,5] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [3,2,4,1,6,5,7] => [3,2,7,1,6,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [2,4,3,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [3,4,2,1,6,5,7] => [3,7,2,1,6,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [3,2,4,5,6,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [3,4,2,5,6,1,7] => [3,7,2,6,5,1,4] => ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [3,2,5,4,6,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [3,4,5,2,6,1,7] => [3,7,6,2,5,1,4] => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [4,3,5,2,6,1,7] => [4,3,7,2,6,1,5] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [2,5,4,3,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [3,5,4,2,6,1,7] => [3,7,6,2,5,1,4] => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [4,5,3,2,6,1,7] => [4,7,3,2,6,1,5] => ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => [5,4,3,2,7,1,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => [2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [3,2,4,6,5,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => ? = 2
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000834
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => [2,1,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => [2,1,5,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => [3,2,5,1,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => [2,5,4,1,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => [3,5,2,1,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,5,4,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => [3,2,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [4,3,2,5,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => [3,2,7,1,6,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => [3,7,2,1,6,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [3,2,4,5,1,6,7] => [3,2,7,6,1,5,4] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [2,4,3,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => [3,7,2,6,1,5,4] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [4,3,2,5,1,6,7] => [4,3,2,7,1,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => [3,2,7,6,1,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [3,4,5,2,1,6,7] => [3,7,6,2,1,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [4,3,5,2,1,6,7] => [4,3,7,2,1,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [2,5,4,3,1,6,7] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [3,5,4,2,1,6,7] => [3,7,6,2,1,5,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => [4,7,3,2,1,6,5] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [3,2,1,4,6,5,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [3,2,4,1,6,5,7] => [3,2,7,1,6,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [2,4,3,1,6,5,7] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [3,4,2,1,6,5,7] => [3,7,2,1,6,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [3,2,4,5,6,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [3,4,2,5,6,1,7] => [3,7,2,6,5,1,4] => ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [3,2,5,4,6,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [3,4,5,2,6,1,7] => [3,7,6,2,5,1,4] => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [4,3,5,2,6,1,7] => [4,3,7,2,6,1,5] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [2,5,4,3,6,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [3,5,4,2,6,1,7] => [3,7,6,2,5,1,4] => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [4,5,3,2,6,1,7] => [4,7,3,2,6,1,5] => ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => [5,4,3,2,7,1,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => [2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [3,2,4,6,5,1,7] => [3,2,7,6,5,1,4] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => ? = 2
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St001011
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 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]
=> [1,0,1,1,0,1,0,0]
=> 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]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 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]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 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]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 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]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[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]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[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]
=> [1,0,1,1,0,1,0,1,0,0]
=> 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]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[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]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[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]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[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]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[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]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[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]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[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]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[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]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[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]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[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]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[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]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[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]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[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]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[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]
=> [1,1,0,1,0,0,1,1,0,0]
=> 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]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[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]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 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]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 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]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 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]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
[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]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[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]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 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]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000386The number of factors DDU in a Dyck path. St001737The number of descents of type 2 in a permutation. St001665The number of pure excedances of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000260The radius of a connected graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000007The number of saliances of the permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001722The number of minimal chains with small intervals between a binary word and the top element. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
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