Processing math: 100%

Your data matches 96 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000237
(load all 6 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => 0
{{1,2}}
 => [2,1] => 1
{{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => 2
{{1,2},{3}}
 => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => 3
{{1,2,3},{4}}
 => [2,3,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 4
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1
Description
The number of small exceedances. This is the number of indices i such that πi=i+1.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => 1 = 0 + 1
{{1,2}}
 => [2] => 2 = 1 + 1
{{1},{2}}
 => [1,1] => 1 = 0 + 1
{{1,2,3}}
 => [3] => 3 = 2 + 1
{{1,2},{3}}
 => [2,1] => 2 = 1 + 1
{{1,3},{2}}
 => [2,1] => 2 = 1 + 1
{{1},{2,3}}
 => [1,2] => 1 = 0 + 1
{{1},{2},{3}}
 => [1,1,1] => 1 = 0 + 1
{{1,2,3,4}}
 => [4] => 4 = 3 + 1
{{1,2,3},{4}}
 => [3,1] => 3 = 2 + 1
{{1,2,4},{3}}
 => [3,1] => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,2] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,1] => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,1] => 3 = 2 + 1
{{1,3},{2,4}}
 => [2,2] => 2 = 1 + 1
{{1,3},{2},{4}}
 => [2,1,1] => 2 = 1 + 1
{{1,4},{2,3}}
 => [2,2] => 2 = 1 + 1
{{1},{2,3,4}}
 => [1,3] => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,2,1] => 1 = 0 + 1
{{1,4},{2},{3}}
 => [2,1,1] => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,2,1] => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,1,2] => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => 1 = 0 + 1
{{1,2,3,4,5}}
 => [5] => 5 = 4 + 1
{{1,2,3,4},{5}}
 => [4,1] => 4 = 3 + 1
{{1,2,3,5},{4}}
 => [4,1] => 4 = 3 + 1
{{1,2,3},{4,5}}
 => [3,2] => 3 = 2 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [4,1] => 4 = 3 + 1
{{1,2,4},{3,5}}
 => [3,2] => 3 = 2 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [3,2] => 3 = 2 + 1
{{1,2},{3,4,5}}
 => [2,3] => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => 3 = 2 + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [4,1] => 4 = 3 + 1
{{1,3,4},{2,5}}
 => [3,2] => 3 = 2 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => 3 = 2 + 1
{{1,3,5},{2,4}}
 => [3,2] => 3 = 2 + 1
{{1,3},{2,4,5}}
 => [2,3] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => 3 = 2 + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [3,2] => 3 = 2 + 1
{{1,4},{2,3,5}}
 => [2,3] => 2 = 1 + 1
Description
The first part of an integer composition.
Matching statistic: St000441
(load all 4 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000441: 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] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 2
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 3
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => 3
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => 1
Description
The number of successions of a permutation. A succession of a permutation π is an index i such that π(i)+1=π(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St001640
(load all 11 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001640: 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] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 2
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 3
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => 3
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => 3
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => 3
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000007
(load all 11 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: 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] => [2,1] => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => 3 = 2 + 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => 2 = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 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] => [1,2,3,5,4] => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => 3 = 2 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => 2 = 1 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => 3 = 2 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => 4 = 3 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => 3 = 2 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => 3 = 2 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => 3 = 2 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => 2 = 1 + 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern ([1],(1,1)), i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000025
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => 1 = 0 + 1
{{1,2}}
 => [2] => [1,1,0,0]
 => 2 = 1 + 1
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of D.
Matching statistic: St000026
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => 1 = 0 + 1
{{1,2}}
 => [2] => [1,1,0,0]
 => 2 = 1 + 1
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000314
(load all 4 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000314: 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] => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,3,1] => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,4,1] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,1,3] => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,4,2,3] => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,3,1,4] => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,1,3,2] => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,4,2,1] => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,4,1] => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,5,1] => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,3,5,2,4] => 3 = 2 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,1,5] => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,3,4,2,1] => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,5,1,4] => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,4,1,5] => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,2,1,4,3] => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [4,3,5,2,1] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,5,1] => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,5,3,4,2] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,1,3,5] => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,1,2,3] => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,5,1,2,3] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,4,2,3,5] => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,3,2,4,1] => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [2,5,3,4,1] => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,5,1] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,1,4] => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,5,3,2,4] => 2 = 1 + 1
Description
The number of left-to-right-maxima of a permutation. An integer σi in the one-line notation of a permutation σ is a '''left-to-right-maximum''' if there does not exist a j<i such that σj>σi. This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000383
(load all 6 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ 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] => [2] => 2 = 1 + 1
{{1},{2}}
 => [1,1] => [1,1] => 1 = 0 + 1
{{1,2,3}}
 => [3] => [3] => 3 = 2 + 1
{{1,2},{3}}
 => [2,1] => [1,2] => 2 = 1 + 1
{{1,3},{2}}
 => [2,1] => [1,2] => 2 = 1 + 1
{{1},{2,3}}
 => [1,2] => [2,1] => 1 = 0 + 1
{{1},{2},{3}}
 => [1,1,1] => [1,1,1] => 1 = 0 + 1
{{1,2,3,4}}
 => [4] => [4] => 4 = 3 + 1
{{1,2,3},{4}}
 => [3,1] => [1,3] => 3 = 2 + 1
{{1,2,4},{3}}
 => [3,1] => [1,3] => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,2] => [2,2] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,2] => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,1] => [1,3] => 3 = 2 + 1
{{1,3},{2,4}}
 => [2,2] => [2,2] => 2 = 1 + 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,2] => 2 = 1 + 1
{{1,4},{2,3}}
 => [2,2] => [2,2] => 2 = 1 + 1
{{1},{2,3,4}}
 => [1,3] => [3,1] => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,2,1] => 1 = 0 + 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,2] => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,2,1] => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,1,2] => [2,1,1] => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
{{1,2,3,4,5}}
 => [5] => [5] => 5 = 4 + 1
{{1,2,3,4},{5}}
 => [4,1] => [1,4] => 4 = 3 + 1
{{1,2,3,5},{4}}
 => [4,1] => [1,4] => 4 = 3 + 1
{{1,2,3},{4,5}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [4,1] => [1,4] => 4 = 3 + 1
{{1,2,4},{3,5}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,2},{3,4,5}}
 => [2,3] => [3,2] => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [2,1,2] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,1,2] => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [4,1] => [1,4] => 4 = 3 + 1
{{1,3,4},{2,5}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,3,5},{2,4}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,3},{2,4,5}}
 => [2,3] => [3,2] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => 3 = 2 + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [2,1,2] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,1,2] => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [3,2] => [2,3] => 3 = 2 + 1
{{1,4},{2,3,5}}
 => [2,3] => [3,2] => 2 = 1 + 1
Description
The last part of an integer composition.
Matching statistic: St000542
(load all 7 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000542: 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] => [2,1] => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 1 = 0 + 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => 3 = 2 + 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => 1 = 0 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => 1 = 0 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => 1 = 0 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 2 = 1 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => 4 = 3 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 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] => [1,2,3,5,4] => 1 = 0 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => 1 = 0 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => 1 = 0 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => 1 = 0 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => 1 = 0 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => 1 = 0 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => 1 = 0 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => 2 = 1 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => 2 = 1 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => 3 = 2 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => 3 = 2 + 1
Description
The number of left-to-right-minima of a permutation. An integer σi in the one-line notation of a permutation σ is a left-to-right-minimum if there does not exist a j < i such that σj<σi.
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000733The row containing the largest entry of a standard tableau. St000439The position of the first down step of a Dyck path. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000214The number of adjacencies of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000445The number of rises of length 1 of a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001484The number of singletons of an integer partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000335The difference of lower and upper interactions. St000363The number of minimal vertex covers of a graph. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000740The last entry of a permutation. St000916The packing number of a graph. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000738The first entry in the last row of a standard tableau. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000932The number of occurrences of the pattern UDU in a Dyck path. St000989The number of final rises of a permutation. St000061The number of nodes on the left branch of a binary tree. St000678The number of up steps after the last double rise of a Dyck path. St000823The number of unsplittable factors of the set partition. St000990The first ascent of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000068The number of minimal elements in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000260The radius of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St000338The number of pixed points of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000654The first descent of a permutation. St001948The number of augmented double ascents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000942The number of critical left to right maxima of the parking functions. St001621The number of atoms of a lattice.