Your data matches 70 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000052
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => [1,0]
 => 0
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 0
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,3,4},{2},{5}}
 => [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]
 => 0
{{1,3,5},{2,4}}
 => [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]
 => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000223
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [1,3,2] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [3,2,1] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [2,3,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [3,4,1,2] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [3,1,4,2] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [4,1,2,3] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [1,4,2,3] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [2,1,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [3,4,2,1] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [3,4,5,1,2] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [3,4,1,5,2] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [4,5,1,2,3] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,4,5,2] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [4,1,5,2,3] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [3,5,1,4,2] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [4,1,2,5,3] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [5,2,1,3,4] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,4,5,3,2] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,2] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [5,1,2,3,4] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [1,5,3,4,2] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,2] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [1,4,5,2,3] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [5,3,1,4,2] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [1,4,2,5,3] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [4,1,5,3,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [4,1,3,5,2] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [1,5,2,3,4] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [5,1,3,4,2] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [1,2,5,4,3] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [1,2,4,5,3] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [2,1,5,3,4] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [4,5,1,3,2] => 1
Description
The number of nestings in the permutation.
Matching statistic: St000356
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,1,3] => [3,2,4,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => [4,3,2,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [3,4,2,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,1,2,4,3] => [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,1,3,2,4] => [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,5,2,3] => [2,5,4,3,1] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,1,3,2,5] => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,1,4,3,2] => [2,5,4,3,1] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,4,5,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,1,3,4,2] => [2,4,5,3,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [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] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,1,3,4] => [3,2,4,5,1] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,5,1,3,2] => [5,3,4,2,1] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,1,3,5] => [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,1,2,3] => [3,4,5,2,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,5,1,2,4] => [4,3,5,1,2] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,1,4,3] => [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => [5,3,2,4,1] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,1,4] => [4,3,2,5,1] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,2,1,3] => [3,4,5,2,1] => 0
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000358
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [3,1,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => [4,2,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => [3,4,1,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [2,3,5,4,1] => [5,2,3,4,1] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,1,5,4] => [2,3,5,1,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,4,3,5,1] => [4,2,3,5,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,5,4,1,2] => [5,3,1,4,2] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [2,4,3,1,5] => [4,2,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,5,4,3,1] => [5,4,2,3,1] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => [2,4,5,1,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,4,5,3,1] => [4,5,2,3,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => 3
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => [3,2,4,5,1] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,4,5,2,1] => [3,4,5,2,1] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [3,4,5,2,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,3,2,5,1] => [4,3,2,5,1] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,5,4,1] => [3,5,2,4,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,3,2,4,1] => [3,2,5,4,1] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,5,1,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,5,1] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,3,4,2,1] => [3,5,4,2,1] => 0
Description
The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St000371
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [1,3,2] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [3,2,1] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [2,3,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [3,4,1,2] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [3,1,4,2] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [4,1,2,3] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [1,4,2,3] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [2,1,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [3,4,2,1] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [3,4,5,1,2] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [3,4,1,5,2] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [4,5,1,2,3] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,4,5,2] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [4,1,5,2,3] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [3,5,1,4,2] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [4,1,2,5,3] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [5,2,1,3,4] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,4,5,3,2] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,2] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [5,1,2,3,4] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [1,5,3,4,2] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,2] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [1,4,5,2,3] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [5,3,1,4,2] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [1,4,2,5,3] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [4,1,5,3,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [4,1,3,5,2] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [1,5,2,3,4] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [5,1,3,4,2] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [1,2,5,4,3] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [1,2,4,5,3] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [2,1,5,3,4] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [4,5,1,3,2] => 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].
Matching statistic: St000663
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,3,2] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => [2,3,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,4,3,2] => [4,3,1,2] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [4,1,3,2] => [4,2,3,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,4,2] => [2,4,1,3] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [3,4,1,2] => [3,1,4,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,4,2,3] => [3,4,1,2] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,4,3] => [1,4,2,3] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,3,1] => [4,1,3,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,2,3,1] => [2,4,3,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,2,4] => [3,1,2,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,5,4,3,2] => [5,4,3,1,2] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,1,4,3,2] => [5,4,2,3,1] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,4,5,3,2] => [2,5,4,1,3] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,5,1,3,2] => [5,3,1,4,2] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [5,4,1,3,2] => [5,3,4,2,1] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,5,3,4,2] => [3,5,4,1,2] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,1,5,4,2] => [1,5,4,2,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [5,1,3,4,2] => [3,5,2,4,1] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,3,4,5,2] => [2,3,5,1,4] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,5,4,1,2] => [1,4,5,3,2] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [5,3,4,1,2] => [4,2,5,3,1] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,4,3,5,2] => [3,2,5,1,4] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,4,5,1,2] => [1,4,2,5,3] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [4,5,3,1,2] => [4,1,5,3,2] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,5,4,2,3] => [4,5,3,1,2] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [2,1,4,5,3] => [1,3,5,2,4] => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [5,1,4,2,3] => [4,5,2,3,1] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,2,5,4,3] => [5,4,1,2,3] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [2,5,1,4,3] => [5,1,3,4,2] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [5,2,1,4,3] => [2,5,3,4,1] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,4,5,2,3] => [4,2,5,1,3] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [2,4,1,5,3] => [5,1,3,2,4] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,5,1,2,3] => [3,4,1,5,2] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,5,2,3,4] => [3,4,5,1,2] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [2,1,5,3,4] => [1,4,5,2,3] => 0
Description
The number of right floats of a permutation. Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of $\pi$. See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St001091
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ 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] => [1,1,0,0]
 => []
 => 0
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1]
 => 0
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 0
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 0
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => []
 => 0
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 0
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001513
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => [2,1] => 0
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [2,3,1] => 0
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [3,1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [4,3,1,2] => 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,4,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [2,3,4,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => 2
Description
The number of nested exceedences of a permutation. For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St001683
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [2,3,1] => 0
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [3,2,4,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,3,1] => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [4,2,1,3] => [2,4,1,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [2,3,4,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,4,2] => [3,1,4,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [3,4,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,4,2] => [1,3,4,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,3,1] => [4,2,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => [3,4,1,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => [1,4,2,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => [3,2,4,5,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => [3,2,4,1,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,3,4,1] => [2,5,3,4,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [5,2,3,1,4] => [2,5,3,1,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [4,2,3,5,1] => [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,1,4,5,3] => [2,1,5,4,3] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,2,3,1] => [4,5,2,3,1] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [4,2,1,5,3] => [2,4,1,5,3] => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,3,4,1] => [5,2,3,4,1] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [5,4,2,1,3] => [4,5,2,1,3] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [5,2,1,3,4] => [2,5,1,3,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => [2,3,4,5,1] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,5,3,4,2] => [5,1,3,4,2] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => [2,3,4,1,5] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [4,3,5,2,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,4,5,1,2] => [4,3,5,1,2] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,3,2,4,1] => [3,5,2,4,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => [5,3,1,4,2] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [5,3,2,1,4] => [3,5,2,1,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,2,5,1] => [3,4,2,5,1] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,3,5,2] => [4,1,3,5,2] => 2
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St001687
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,1,2] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => [4,2,1,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [4,3,2,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => [4,1,2,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,4,2,1] => [4,1,3,2] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => [5,2,3,4,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [2,3,5,4,1] => [5,2,3,1,4] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,4,3,5,1] => [5,2,4,3,1] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,5,4,1,2] => [2,5,3,1,4] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [2,4,3,1,5] => [4,2,1,3,5] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,5,4,3,1] => [5,2,1,3,4] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,4,5,3,1] => [5,2,1,4,3] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,3,4] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [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] => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => [5,3,2,4,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,4,5,2,1] => [5,1,3,4,2] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => [4,3,2,1,5] => 3
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [5,1,3,4,2] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,3,2,5,1] => [5,4,2,3,1] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,1,2,3,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,5,4,1] => [5,3,2,1,4] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,3,2,4,1] => [5,4,2,1,3] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [3,1,2,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,1,2,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,2,5,1] => [5,4,2,3,1] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,3,4,2,1] => [5,1,4,2,3] => 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000941The number of characters of the symmetric group whose value on the partition is even. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001820The size of the image of the pop stack sorting operator. St001651The Frankl number of a lattice. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001435The number of missing boxes in the first row. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001857The number of edges in the reduced word graph of a signed permutation.