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Your data matches 89 different statistics following compositions of up to 3 maps.
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Matching statistic: St000021
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(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000023
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(load all 12 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000035
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(load all 6 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 1
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: St000884
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(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 1
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001037
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001729
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001928
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 1
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000092
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1 = 0 + 1
{{1,2}}
=> [2,1] => [2,1] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,4,3,2,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,1,5] => 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,3,2,1] => 1 = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,2,1] => 1 = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,2,1,5] => 2 = 1 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,2,1] => 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,2,1] => 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,5,3,2] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,3,1,5] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,4,1,3,2] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,3,2,5] => 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => 2 = 1 + 1
Description
The number of outer peaks of a permutation.
An 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 $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000099
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1 = 0 + 1
{{1,2}}
=> [2,1] => [1,2] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 2 = 1 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2 = 1 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 2 = 1 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 2 = 1 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 2 = 1 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2 = 1 + 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000325
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1 = 0 + 1
{{1,2}}
=> [2,1] => [1,2] => 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 2 = 1 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2 = 1 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 2 = 1 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 2 = 1 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 2 = 1 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2 = 1 + 1
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
The following 79 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000470The number of runs in a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000245The number of ascents of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000386The number of factors DDU in a Dyck path. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001214The aft of an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000054The first entry of the permutation. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000159The number of distinct parts of the integer partition. St000201The number of leaf nodes in a binary tree. St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000935The number of ordered refinements of an integer partition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001732The number of peaks visible from the left. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000354The number of recoils of a permutation. St000389The number of runs of ones of odd length in a binary word. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000919The number of maximal left branches of a binary tree. St001984A recursive count of subwords of the form 01, 10 and 11. St000568The hook number of a binary tree. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001427The number of descents of a signed permutation. St001354The number of series nodes in the modular decomposition of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001128The exponens consonantiae of a partition. St000454The largest eigenvalue of a graph if it is integral. St001597The Frobenius rank of a skew partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001487The number of inner corners of a skew partition. St001737The number of descents of type 2 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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