Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001075
(load all 12 compositions to match this statistic)
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => 2
{{1},{2}}
 => {{1},{2}}
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => 1
Description
The minimal size of a block of a set partition.
Matching statistic: St000657
(load all 9 compositions to match this statistic)
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [2] => 2
{{1},{2}}
 => {{1},{2}}
 => [1,1] => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [1,3] => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2] => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [1,4] => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [2,3] => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2] => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000655
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [2] => [1,1,0,0]
 => 2
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
{{1},{2,3},{4,5,6,7,8}}
 => {{1,3,5,6,7,8},{2},{4}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
Description
The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].
Matching statistic: St000993
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [2]
 => [1,1]
 => 2
{{1},{2}}
 => {{1},{2}}
 => [1,1]
 => [2]
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3]
 => [1,1,1]
 => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1]
 => [2,1]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [2,1]
 => [2,1]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1]
 => [2,1]
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1]
 => [3]
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4]
 => [1,1,1,1]
 => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1]
 => [2,1,1]
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2]
 => [2,2]
 => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1]
 => [2,1,1]
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1]
 => [3,1]
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [3,1]
 => [2,1,1]
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2]
 => [2,2]
 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [2,1,1]
 => [3,1]
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2]
 => [2,2]
 => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1]
 => [2,1,1]
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1]
 => [3,1]
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1]
 => [4]
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1]
 => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1]
 => [2,1,1,1]
 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1]
 => [2,1,1,1]
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1]
 => [3,1,1]
 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1]
 => [3,2]
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1]
 => [2,1,1,1]
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1]
 => [3,1,1]
 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,2,1]
 => [3,2]
 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1]
 => [3,2]
 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1]
 => [3,1,1]
 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [4,1]
 => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [4,1]
 => [2,1,1,1]
 => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [3,1,1]
 => [3,1,1]
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1]
 => [3,2]
 => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1]
 => [3,2]
 => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [2,2,1]
 => [3,2]
 => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [3,1,1]
 => [3,1,1]
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [4,1]
 => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [3,2]
 => [2,2,1]
 => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1]
 => [3,2]
 => 1
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => ?
 => ?
 => ? = 1
{{1},{2,3},{4,5,6,7,8}}
 => {{1,3,5,6,7,8},{2},{4}}
 => ?
 => ?
 => ? = 1
{{1,3},{2},{4,5,6,7,8}}
 => {{1},{2,3,5,6,7,8},{4}}
 => ?
 => ?
 => ? = 1
{{1,3,4,5,6,7},{2},{8}}
 => {{1},{2,3,4,5,6,7},{8}}
 => ?
 => ?
 => ? = 1
{{1,2,4,5,6,7,8},{3}}
 => {{1,2,5,6,7,8},{3,4}}
 => ?
 => ?
 => ? = 2
{{1,6,7,8},{2,3,4,5}}
 => {{1,3,4,5,7,8},{2,6}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,6,7,8},{5}}
 => {{1,2,3,4,7,8},{5,6}}
 => ?
 => ?
 => ? = 2
{{1,7,8},{2,3,4,5,6}}
 => {{1,3,4,5,6,8},{2,7}}
 => ?
 => ?
 => ? = 2
{{1,8},{2,3,4,5,6,7}}
 => {{1,3,4,5,6,7},{2,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,8},{4,5,6,7}}
 => {{1,2,3,5,6,7},{4,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,8},{5,6,7}}
 => {{1,2,3,4,6,7},{5,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,5,8},{6,7}}
 => {{1,2,3,4,5,7},{6,8}}
 => ?
 => ?
 => ? = 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => [2]
 => [1,0,1,0]
 => 2
{{1},{2}}
 => {{1},{2}}
 => [1,1]
 => [1,1,0,0]
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3]
 => [1,0,1,0,1,0]
 => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => ?
 => ?
 => ? = 1
{{1},{2,3},{4,5,6,7,8}}
 => {{1,3,5,6,7,8},{2},{4}}
 => ?
 => ?
 => ? = 1
{{1,3},{2},{4,5,6,7,8}}
 => {{1},{2,3,5,6,7,8},{4}}
 => ?
 => ?
 => ? = 1
{{1,3,4,5,6,7},{2},{8}}
 => {{1},{2,3,4,5,6,7},{8}}
 => ?
 => ?
 => ? = 1
{{1,2,4,5,6,7,8},{3}}
 => {{1,2,5,6,7,8},{3,4}}
 => ?
 => ?
 => ? = 2
{{1,6,7,8},{2,3,4,5}}
 => {{1,3,4,5,7,8},{2,6}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,6,7,8},{5}}
 => {{1,2,3,4,7,8},{5,6}}
 => ?
 => ?
 => ? = 2
{{1,7,8},{2,3,4,5,6}}
 => {{1,3,4,5,6,8},{2,7}}
 => ?
 => ?
 => ? = 2
{{1,8},{2,3,4,5,6,7}}
 => {{1,3,4,5,6,7},{2,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,8},{4,5,6,7}}
 => {{1,2,3,5,6,7},{4,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,8},{5,6,7}}
 => {{1,2,3,4,6,7},{5,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,5,8},{6,7}}
 => {{1,2,3,4,5,7},{6,8}}
 => ?
 => ?
 => ? = 2
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000667
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 98%●distinct values known / distinct values provided: 38%
Values
{{1,2}}
 => {{1,2}}
 => [2]
 => []
 => ? = 2
{{1},{2}}
 => {{1},{2}}
 => [1,1]
 => [1]
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3]
 => []
 => ? = 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1]
 => [1]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [2,1]
 => [1]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1]
 => [1]
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1]
 => [1,1]
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4]
 => []
 => ? = 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1]
 => [1]
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2]
 => [2]
 => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1]
 => [1]
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1]
 => [1,1]
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [3,1]
 => [1]
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2]
 => [2]
 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [2,1,1]
 => [1,1]
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2]
 => [2]
 => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1]
 => [1]
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1]
 => [1,1]
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5]
 => []
 => ? = 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1]
 => [1]
 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2]
 => [2]
 => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1]
 => [1]
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,1]
 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2]
 => [2]
 => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [3,2]
 => [2]
 => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2]
 => [2]
 => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1]
 => [1]
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,1]
 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [4,1]
 => [1]
 => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [3,2]
 => [2]
 => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,1]
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [3,2]
 => [2]
 => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2]
 => [2]
 => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2]
 => [2]
 => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [3,2]
 => [2]
 => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,5},{2,3,4}}
 => {{1,3,4},{2,5}}
 => [3,2]
 => [2]
 => 2
{{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => [4,1]
 => [1]
 => 1
{{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,1]
 => 1
{{1,5},{2,3},{4}}
 => {{1,3},{2},{4,5}}
 => [2,2,1]
 => [2,1]
 => 1
{{1,2,3,4,5,6}}
 => {{1,2,3,4,5,6}}
 => [6]
 => []
 => ? = 6
{{1,2,3,4,5,6,7}}
 => {{1,2,3,4,5,6,7}}
 => [7]
 => []
 => ? = 7
{{1},{2},{3,4,5,6,7,8}}
 => {{1,4,5,6,7,8},{2},{3}}
 => ?
 => ?
 => ? = 1
{{1},{2,3},{4,5,6,7,8}}
 => {{1,3,5,6,7,8},{2},{4}}
 => ?
 => ?
 => ? = 1
{{1,3},{2},{4,5,6,7,8}}
 => {{1},{2,3,5,6,7,8},{4}}
 => ?
 => ?
 => ? = 1
{{1,3,4,5,6,7},{2},{8}}
 => {{1},{2,3,4,5,6,7},{8}}
 => ?
 => ?
 => ? = 1
{{1,2,4,5,6,7,8},{3}}
 => {{1,2,5,6,7,8},{3,4}}
 => ?
 => ?
 => ? = 2
{{1,6,7,8},{2,3,4,5}}
 => {{1,3,4,5,7,8},{2,6}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,6,7,8},{5}}
 => {{1,2,3,4,7,8},{5,6}}
 => ?
 => ?
 => ? = 2
{{1,7,8},{2,3,4,5,6}}
 => {{1,3,4,5,6,8},{2,7}}
 => ?
 => ?
 => ? = 2
{{1,8},{2,3,4,5,6,7}}
 => {{1,3,4,5,6,7},{2,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,8},{4,5,6,7}}
 => {{1,2,3,5,6,7},{4,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,8},{5,6,7}}
 => {{1,2,3,4,6,7},{5,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,5,8},{6,7}}
 => {{1,2,3,4,5,7},{6,8}}
 => ?
 => ?
 => ? = 2
{{1,2,3,4,5,6,7,8}}
 => {{1,2,3,4,5,6,7,8}}
 => [8]
 => []
 => ? = 8
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000260
(load all 5 compositions to match this statistic)
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 57%●distinct values known / distinct values provided: 12%
Values
{{1,2}}
 => {{1,2}}
 => [2] => ([],2)
 => ? = 2
{{1},{2}}
 => {{1},{2}}
 => [1,1] => ([(0,1)],2)
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => ([],3)
 => ? = 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => ([(1,2)],3)
 => ? = 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => ([],4)
 => ? = 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [1,3] => ([(2,3)],4)
 => ? = 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => ([],5)
 => ? = 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [1,4] => ([(3,4)],5)
 => ? = 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,5},{2,3,4}}
 => {{1,3,4},{2,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
{{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,5},{2,3},{4}}
 => {{1,3},{2},{4,5}}
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1},{2,3,5},{4}}
 => {{1,3},{2,5},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,4,5},{2},{3}}
 => {{1},{2},{3,4,5}}
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1,4},{2,5},{3}}
 => {{1},{2,5},{3,4}}
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1,4},{2},{3,5}}
 => {{1,5},{2},{3,4}}
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1,4},{2},{3},{5}}
 => {{1},{2},{3,4},{5}}
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,5},{2,4},{3}}
 => {{1},{2,4},{3,5}}
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1},{2,4,5},{3}}
 => {{1},{2,4,5},{3}}
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2,4},{3,5}}
 => {{1,5},{2,4},{3}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,5},{2},{3,4}}
 => {{1,4},{2},{3,5}}
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1},{2,5},{3,4}}
 => {{1,4},{2,5},{3}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,5},{2},{3},{4}}
 => {{1},{2},{3},{4,5}}
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
{{1},{2,5},{3},{4}}
 => {{1},{2},{3,5},{4}}
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2},{3,5},{4}}
 => {{1},{2,5},{3},{4}}
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,3,4,5,6}}
 => {{1,2,3,4,5,6}}
 => [6] => ([],6)
 => ? = 6
{{1,2,3,4,5},{6}}
 => {{1,2,3,4,5},{6}}
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1,2,3,4,6},{5}}
 => {{1,2,3,4},{5,6}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
{{1,2,3,4},{5,6}}
 => {{1,2,3,4,6},{5}}
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1,2,3,4},{5},{6}}
 => {{1,2,3,4},{5},{6}}
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,3,5,6},{4}}
 => {{1,2,3,6},{4,5}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
{{1,2,3,5},{4,6}}
 => {{1,2,3},{4,5,6}}
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 3
{{1,2,3,5},{4},{6}}
 => {{1,2,3},{4,5},{6}}
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,3,6},{4,5}}
 => {{1,2,3,5},{4,6}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
{{1,2,3},{4,5,6}}
 => {{1,2,3,5,6},{4}}
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
{{1,2,3},{4,5},{6}}
 => {{1,2,3,5},{4},{6}}
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,3,6},{4},{5}}
 => {{1,2,3},{4},{5,6}}
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1,2,4,5,6},{3}}
 => {{1,2,5,6},{3,4}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
{{1,2,4,5},{3,6}}
 => {{1,2,5},{3,4,6}}
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 3
{{1,2,4,6},{3,5}}
 => {{1,2,6},{3,4,5}}
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 3
{{1,2,4},{3,5,6}}
 => {{1,2},{3,4,5,6}}
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 2
{{1,2,4},{3,6},{5}}
 => {{1,2},{3,4},{5,6}}
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
{{1,2,5,6},{3,4}}
 => {{1,2,4,6},{3,5}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
{{1,2,5},{3,4,6}}
 => {{1,2,4},{3,5,6}}
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 3
{{1,2,6},{3,4,5}}
 => {{1,2,4,5},{3,6}}
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
{{1,2,6},{3,4},{5}}
 => {{1,2,4},{3},{5,6}}
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1,2,5,6},{3},{4}}
 => {{1,2,6},{3},{4,5}}
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1,2,5},{3,6},{4}}
 => {{1,2},{3,6},{4,5}}
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
{{1,2,5},{3},{4,6}}
 => {{1,2},{3},{4,5,6}}
 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
{{1,2,6},{3,5},{4}}
 => {{1,2},{3,5},{4,6}}
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
{{1,2,6},{3},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000487
(load all 5 compositions to match this statistic)
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000487: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 75%
Values
{{1,2}}
 => {{1,2}}
 => [2,1] => 2
{{1},{2}}
 => {{1},{2}}
 => [1,2] => 1
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,1,4,5,3] => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [4,3,2,5,1] => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [5,3,2,4,1] => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1
{{1,2,3,4,5,6,7}}
 => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 7
{{1,2,3,4,5,6},{7}}
 => {{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 1
{{1,2,3,4,5,7},{6}}
 => {{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => ? = 2
{{1,2,3,4,5},{6,7}}
 => {{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => ? = 1
{{1,2,3,4,5},{6},{7}}
 => {{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => ? = 1
{{1,2,3,4,6,7},{5}}
 => {{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => ? = 2
{{1,2,3,4,6},{5,7}}
 => {{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => ? = 3
{{1,2,3,4,6},{5},{7}}
 => {{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => ? = 1
{{1,2,3,4,7},{5,6}}
 => {{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => ? = 2
{{1,2,3,4},{5,6,7}}
 => {{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => ? = 1
{{1,2,3,4},{5,6},{7}}
 => {{1,2,3,4,6},{5},{7}}
 => [2,3,4,6,5,1,7] => ? = 1
{{1,2,3,4,7},{5},{6}}
 => {{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => ? = 1
{{1,2,3,4},{5,7},{6}}
 => {{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => ? = 1
{{1,2,3,4},{5},{6,7}}
 => {{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => ? = 1
{{1,2,3,4},{5},{6},{7}}
 => {{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => ? = 1
{{1,2,3,5,6,7},{4}}
 => {{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => ? = 2
{{1,2,3,5,6},{4,7}}
 => {{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => ? = 3
{{1,2,3,5,6},{4},{7}}
 => {{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => ? = 1
{{1,2,3,5,7},{4,6}}
 => {{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => ? = 3
{{1,2,3,5},{4,6,7}}
 => {{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => ? = 3
{{1,2,3,5},{4,6},{7}}
 => {{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => ? = 1
{{1,2,3,5,7},{4},{6}}
 => {{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => ? = 1
{{1,2,3,5},{4,7},{6}}
 => {{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => ? = 2
{{1,2,3,5},{4},{6,7}}
 => {{1,2,3,7},{4,5},{6}}
 => [2,3,7,5,4,6,1] => ? = 1
{{1,2,3,5},{4},{6},{7}}
 => {{1,2,3},{4,5},{6},{7}}
 => [2,3,1,5,4,6,7] => ? = 1
{{1,2,3,6,7},{4,5}}
 => {{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => ? = 2
{{1,2,3,6},{4,5,7}}
 => {{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => ? = 3
{{1,2,3,6},{4,5},{7}}
 => {{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => ? = 1
{{1,2,3,7},{4,5,6}}
 => {{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => ? = 2
{{1,2,3},{4,5,6,7}}
 => {{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => ? = 1
{{1,2,3},{4,5,6},{7}}
 => {{1,2,3,5,6},{4},{7}}
 => [2,3,5,4,6,1,7] => ? = 1
{{1,2,3,7},{4,5},{6}}
 => {{1,2,3,5},{4},{6,7}}
 => [2,3,5,4,1,7,6] => ? = 1
{{1,2,3},{4,5,7},{6}}
 => {{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => ? = 1
{{1,2,3},{4,5},{6,7}}
 => {{1,2,3,5,7},{4},{6}}
 => [2,3,5,4,7,6,1] => ? = 1
{{1,2,3},{4,5},{6},{7}}
 => {{1,2,3,5},{4},{6},{7}}
 => [2,3,5,4,1,6,7] => ? = 1
{{1,2,3,6,7},{4},{5}}
 => {{1,2,3,7},{4},{5,6}}
 => [2,3,7,4,6,5,1] => ? = 1
{{1,2,3,6},{4,7},{5}}
 => {{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => ? = 2
{{1,2,3,6},{4},{5,7}}
 => {{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => ? = 1
{{1,2,3,6},{4},{5},{7}}
 => {{1,2,3},{4},{5,6},{7}}
 => [2,3,1,4,6,5,7] => ? = 1
{{1,2,3,7},{4,6},{5}}
 => {{1,2,3},{4,6},{5,7}}
 => [2,3,1,6,7,4,5] => ? = 2
{{1,2,3},{4,6,7},{5}}
 => {{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => ? = 1
{{1,2,3},{4,6},{5,7}}
 => {{1,2,3},{4,6,7},{5}}
 => [2,3,1,6,5,7,4] => ? = 1
{{1,2,3},{4,6},{5},{7}}
 => {{1,2,3},{4,6},{5},{7}}
 => [2,3,1,6,5,4,7] => ? = 1
{{1,2,3,7},{4},{5,6}}
 => {{1,2,3,6},{4},{5,7}}
 => [2,3,6,4,7,1,5] => ? = 1
{{1,2,3},{4,7},{5,6}}
 => {{1,2,3,6},{4,7},{5}}
 => [2,3,6,7,5,1,4] => ? = 1
{{1,2,3},{4},{5,6,7}}
 => {{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => ? = 1
{{1,2,3},{4},{5,6},{7}}
 => {{1,2,3,6},{4},{5},{7}}
 => [2,3,6,4,5,1,7] => ? = 1
{{1,2,3,7},{4},{5},{6}}
 => {{1,2,3},{4},{5},{6,7}}
 => [2,3,1,4,5,7,6] => ? = 1
{{1,2,3},{4,7},{5},{6}}
 => {{1,2,3},{4},{5,7},{6}}
 => [2,3,1,4,7,6,5] => ? = 1
{{1,2,3},{4},{5,7},{6}}
 => {{1,2,3},{4,7},{5},{6}}
 => [2,3,1,7,5,6,4] => ? = 1
Description
The length of the shortest cycle of a permutation.
Matching statistic: St000654
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 88%
Values
{{1,2}}
 => {{1,2}}
 => [[1,2]]
 => [1,2] => 2
{{1},{2}}
 => {{1},{2}}
 => [[1],[2]]
 => [2,1] => 1
{{1,2,3}}
 => {{1,2,3}}
 => [[1,2,3]]
 => [1,2,3] => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [[1,2],[3]]
 => [3,1,2] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [[1,3],[2]]
 => [2,1,3] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [[1,3],[2]]
 => [2,1,3] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [[1],[2],[3]]
 => [3,2,1] => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [[1,2,3,4]]
 => [1,2,3,4] => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [4,1,2,3] => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [3,1,2,4] => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [[1,3,4],[2],[5]]
 => [5,2,1,3,4] => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [[1,3,5],[2],[4]]
 => [4,2,1,3,5] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 1
{{1,2,3,4,5,6},{7}}
 => {{1,2,3,4,5,6},{7}}
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 1
{{1,2,3,4,5,7},{6}}
 => {{1,2,3,4,5},{6,7}}
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 2
{{1,2,3,4,5},{6,7}}
 => {{1,2,3,4,5,7},{6}}
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => ? = 1
{{1,2,3,4,5},{6},{7}}
 => {{1,2,3,4,5},{6},{7}}
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ? = 1
{{1,2,3,4,6,7},{5}}
 => {{1,2,3,4,7},{5,6}}
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => ? = 2
{{1,2,3,4,6},{5,7}}
 => {{1,2,3,4},{5,6,7}}
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ? = 3
{{1,2,3,4,6},{5},{7}}
 => {{1,2,3,4},{5,6},{7}}
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ? = 1
{{1,2,3,4,7},{5,6}}
 => {{1,2,3,4,6},{5,7}}
 => [[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => ? = 2
{{1,2,3,4},{5,6,7}}
 => {{1,2,3,4,6,7},{5}}
 => [[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => ? = 1
{{1,2,3,4},{5,6},{7}}
 => {{1,2,3,4,6},{5},{7}}
 => [[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => ? = 1
{{1,2,3,4,7},{5},{6}}
 => {{1,2,3,4},{5},{6,7}}
 => [[1,2,3,4],[5,7],[6]]
 => [6,5,7,1,2,3,4] => ? = 1
{{1,2,3,4},{5,7},{6}}
 => {{1,2,3,4},{5,7},{6}}
 => [[1,2,3,4],[5,7],[6]]
 => [6,5,7,1,2,3,4] => ? = 1
{{1,2,3,4},{5},{6,7}}
 => {{1,2,3,4,7},{5},{6}}
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => ? = 1
{{1,2,3,4},{5},{6},{7}}
 => {{1,2,3,4},{5},{6},{7}}
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ? = 1
{{1,2,3,5,6,7},{4}}
 => {{1,2,3,6,7},{4,5}}
 => [[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => ? = 2
{{1,2,3,5,6},{4,7}}
 => {{1,2,3,6},{4,5,7}}
 => [[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => ? = 3
{{1,2,3,5,6},{4},{7}}
 => {{1,2,3,6},{4,5},{7}}
 => [[1,2,3,6],[4,5],[7]]
 => [7,4,5,1,2,3,6] => ? = 1
{{1,2,3,5,7},{4,6}}
 => {{1,2,3},{4,5,6,7}}
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 3
{{1,2,3,5},{4,6,7}}
 => {{1,2,3,7},{4,5,6}}
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 3
{{1,2,3,5},{4,6},{7}}
 => {{1,2,3},{4,5,6},{7}}
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ? = 1
{{1,2,3,5,7},{4},{6}}
 => {{1,2,3},{4,5,7},{6}}
 => [[1,2,3],[4,5,7],[6]]
 => [6,4,5,7,1,2,3] => ? = 1
{{1,2,3,5},{4,7},{6}}
 => {{1,2,3},{4,5},{6,7}}
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 2
{{1,2,3,5},{4},{6,7}}
 => {{1,2,3,7},{4,5},{6}}
 => [[1,2,3,7],[4,5],[6]]
 => [6,4,5,1,2,3,7] => ? = 1
{{1,2,3,5},{4},{6},{7}}
 => {{1,2,3},{4,5},{6},{7}}
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ? = 1
{{1,2,3,6,7},{4,5}}
 => {{1,2,3,5,7},{4,6}}
 => [[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => ? = 2
{{1,2,3,6},{4,5,7}}
 => {{1,2,3,5},{4,6,7}}
 => [[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => ? = 3
{{1,2,3,6},{4,5},{7}}
 => {{1,2,3,5},{4,6},{7}}
 => [[1,2,3,5],[4,6],[7]]
 => [7,4,6,1,2,3,5] => ? = 1
{{1,2,3,7},{4,5,6}}
 => {{1,2,3,5,6},{4,7}}
 => [[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => ? = 2
{{1,2,3},{4,5,6,7}}
 => {{1,2,3,5,6,7},{4}}
 => [[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => ? = 1
{{1,2,3},{4,5,6},{7}}
 => {{1,2,3,5,6},{4},{7}}
 => [[1,2,3,5,6],[4],[7]]
 => [7,4,1,2,3,5,6] => ? = 1
{{1,2,3,7},{4,5},{6}}
 => {{1,2,3,5},{4},{6,7}}
 => [[1,2,3,5],[4,7],[6]]
 => [6,4,7,1,2,3,5] => ? = 1
{{1,2,3},{4,5,7},{6}}
 => {{1,2,3,5},{4,7},{6}}
 => [[1,2,3,5],[4,7],[6]]
 => [6,4,7,1,2,3,5] => ? = 1
{{1,2,3},{4,5},{6,7}}
 => {{1,2,3,5,7},{4},{6}}
 => [[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => ? = 1
{{1,2,3},{4,5},{6},{7}}
 => {{1,2,3,5},{4},{6},{7}}
 => [[1,2,3,5],[4],[6],[7]]
 => [7,6,4,1,2,3,5] => ? = 1
{{1,2,3,6,7},{4},{5}}
 => {{1,2,3,7},{4},{5,6}}
 => [[1,2,3,7],[4,6],[5]]
 => [5,4,6,1,2,3,7] => ? = 1
{{1,2,3,6},{4,7},{5}}
 => {{1,2,3},{4,7},{5,6}}
 => [[1,2,3],[4,6],[5,7]]
 => [5,7,4,6,1,2,3] => ? = 2
{{1,2,3,6},{4},{5,7}}
 => {{1,2,3},{4},{5,6,7}}
 => [[1,2,3],[4,6,7],[5]]
 => [5,4,6,7,1,2,3] => ? = 1
{{1,2,3,6},{4},{5},{7}}
 => {{1,2,3},{4},{5,6},{7}}
 => [[1,2,3],[4,6],[5],[7]]
 => [7,5,4,6,1,2,3] => ? = 1
{{1,2,3,7},{4,6},{5}}
 => {{1,2,3},{4,6},{5,7}}
 => [[1,2,3],[4,6],[5,7]]
 => [5,7,4,6,1,2,3] => ? = 2
{{1,2,3},{4,6,7},{5}}
 => {{1,2,3,7},{4,6},{5}}
 => [[1,2,3,7],[4,6],[5]]
 => [5,4,6,1,2,3,7] => ? = 1
{{1,2,3},{4,6},{5,7}}
 => {{1,2,3},{4,6,7},{5}}
 => [[1,2,3],[4,6,7],[5]]
 => [5,4,6,7,1,2,3] => ? = 1
{{1,2,3},{4,6},{5},{7}}
 => {{1,2,3},{4,6},{5},{7}}
 => [[1,2,3],[4,6],[5],[7]]
 => [7,5,4,6,1,2,3] => ? = 1
{{1,2,3,7},{4},{5,6}}
 => {{1,2,3,6},{4},{5,7}}
 => [[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => ? = 1
{{1,2,3},{4,7},{5,6}}
 => {{1,2,3,6},{4,7},{5}}
 => [[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => ? = 1
{{1,2,3},{4},{5,6,7}}
 => {{1,2,3,6,7},{4},{5}}
 => [[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => ? = 1
{{1,2,3},{4},{5,6},{7}}
 => {{1,2,3,6},{4},{5},{7}}
 => [[1,2,3,6],[4],[5],[7]]
 => [7,5,4,1,2,3,6] => ? = 1
{{1,2,3,7},{4},{5},{6}}
 => {{1,2,3},{4},{5},{6,7}}
 => [[1,2,3],[4,7],[5],[6]]
 => [6,5,4,7,1,2,3] => ? = 1
{{1,2,3},{4,7},{5},{6}}
 => {{1,2,3},{4},{5,7},{6}}
 => [[1,2,3],[4,7],[5],[6]]
 => [6,5,4,7,1,2,3] => ? = 1
{{1,2,3},{4},{5,7},{6}}
 => {{1,2,3},{4,7},{5},{6}}
 => [[1,2,3],[4,7],[5],[6]]
 => [6,5,4,7,1,2,3] => ? = 1
{{1,2,3},{4},{5},{6,7}}
 => {{1,2,3,7},{4},{5},{6}}
 => [[1,2,3,7],[4],[5],[6]]
 => [6,5,4,1,2,3,7] => ? = 1
Description
The first descent of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St001236
(load all 2 compositions to match this statistic)
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Values
{{1,2}}
 => {{1,2}}
 => [2] => [1,1] => 2
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [2] => 1
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1] => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [2,1] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,2] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [2,1] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [3] => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1] => 4
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [2,1,1] => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,2,1] => 2
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [2,1,1] => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [3,1] => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [1,3] => [1,1,2] => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2] => [1,2,1] => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,2,1] => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [2,1,1] => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [3,1] => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,3] => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [3,1] => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 5
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,2,1,1] => 2
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,2,1,1] => 2
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,1,2,1] => 2
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,2,1,1] => 2
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [2,1,1,1] => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [3,1,1] => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [4,1] => 1
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => [1,4] => [1,1,1,2] => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,1,2,1] => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [2,1,2] => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,2,1] => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,2,1,1] => 2
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,2,2] => 1
{{1,3},{2},{4,5}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [2,1,2] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [3,2] => 1
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,2,1,1] => 2
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,1,2,1] => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,2,3,4,5,6,7}}
 => {{1,2,3,4,5,6,7}}
 => [7] => [1,1,1,1,1,1,1] => ? = 7
{{1,2,3,4,5,6},{7}}
 => {{1,2,3,4,5,6},{7}}
 => [6,1] => [2,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,7},{6}}
 => {{1,2,3,4,5},{6,7}}
 => [5,2] => [1,2,1,1,1,1] => ? = 2
{{1,2,3,4,5},{6,7}}
 => {{1,2,3,4,5,7},{6}}
 => [6,1] => [2,1,1,1,1,1] => ? = 1
{{1,2,3,4,5},{6},{7}}
 => {{1,2,3,4,5},{6},{7}}
 => [5,1,1] => [3,1,1,1,1] => ? = 1
{{1,2,3,4,6,7},{5}}
 => {{1,2,3,4,7},{5,6}}
 => [5,2] => [1,2,1,1,1,1] => ? = 2
{{1,2,3,4,6},{5,7}}
 => {{1,2,3,4},{5,6,7}}
 => [4,3] => [1,1,2,1,1,1] => ? = 3
{{1,2,3,4,6},{5},{7}}
 => {{1,2,3,4},{5,6},{7}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3,4,7},{5,6}}
 => {{1,2,3,4,6},{5,7}}
 => [5,2] => [1,2,1,1,1,1] => ? = 2
{{1,2,3,4},{5,6,7}}
 => {{1,2,3,4,6,7},{5}}
 => [6,1] => [2,1,1,1,1,1] => ? = 1
{{1,2,3,4},{5,6},{7}}
 => {{1,2,3,4,6},{5},{7}}
 => [5,1,1] => [3,1,1,1,1] => ? = 1
{{1,2,3,4,7},{5},{6}}
 => {{1,2,3,4},{5},{6,7}}
 => [4,1,2] => [1,3,1,1,1] => ? = 1
{{1,2,3,4},{5,7},{6}}
 => {{1,2,3,4},{5,7},{6}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3,4},{5},{6,7}}
 => {{1,2,3,4,7},{5},{6}}
 => [5,1,1] => [3,1,1,1,1] => ? = 1
{{1,2,3,4},{5},{6},{7}}
 => {{1,2,3,4},{5},{6},{7}}
 => [4,1,1,1] => [4,1,1,1] => ? = 1
{{1,2,3,5,6,7},{4}}
 => {{1,2,3,6,7},{4,5}}
 => [5,2] => [1,2,1,1,1,1] => ? = 2
{{1,2,3,5,6},{4,7}}
 => {{1,2,3,6},{4,5,7}}
 => [4,3] => [1,1,2,1,1,1] => ? = 3
{{1,2,3,5,6},{4},{7}}
 => {{1,2,3,6},{4,5},{7}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3,5,7},{4,6}}
 => {{1,2,3},{4,5,6,7}}
 => [3,4] => [1,1,1,2,1,1] => ? = 3
{{1,2,3,5},{4,6,7}}
 => {{1,2,3,7},{4,5,6}}
 => [4,3] => [1,1,2,1,1,1] => ? = 3
{{1,2,3,5},{4,6},{7}}
 => {{1,2,3},{4,5,6},{7}}
 => [3,3,1] => [2,1,2,1,1] => ? = 1
{{1,2,3,5,7},{4},{6}}
 => {{1,2,3},{4,5,7},{6}}
 => [3,3,1] => [2,1,2,1,1] => ? = 1
{{1,2,3,5},{4,7},{6}}
 => {{1,2,3},{4,5},{6,7}}
 => [3,2,2] => [1,2,2,1,1] => ? = 2
{{1,2,3,5},{4},{6,7}}
 => {{1,2,3,7},{4,5},{6}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3,5},{4},{6},{7}}
 => {{1,2,3},{4,5},{6},{7}}
 => [3,2,1,1] => [3,2,1,1] => ? = 1
{{1,2,3,6,7},{4,5}}
 => {{1,2,3,5,7},{4,6}}
 => [5,2] => [1,2,1,1,1,1] => ? = 2
{{1,2,3,6},{4,5,7}}
 => {{1,2,3,5},{4,6,7}}
 => [4,3] => [1,1,2,1,1,1] => ? = 3
{{1,2,3,6},{4,5},{7}}
 => {{1,2,3,5},{4,6},{7}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3,7},{4,5,6}}
 => {{1,2,3,5,6},{4,7}}
 => [5,2] => [1,2,1,1,1,1] => ? = 2
{{1,2,3},{4,5,6,7}}
 => {{1,2,3,5,6,7},{4}}
 => [6,1] => [2,1,1,1,1,1] => ? = 1
{{1,2,3},{4,5,6},{7}}
 => {{1,2,3,5,6},{4},{7}}
 => [5,1,1] => [3,1,1,1,1] => ? = 1
{{1,2,3,7},{4,5},{6}}
 => {{1,2,3,5},{4},{6,7}}
 => [4,1,2] => [1,3,1,1,1] => ? = 1
{{1,2,3},{4,5,7},{6}}
 => {{1,2,3,5},{4,7},{6}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3},{4,5},{6,7}}
 => {{1,2,3,5,7},{4},{6}}
 => [5,1,1] => [3,1,1,1,1] => ? = 1
{{1,2,3},{4,5},{6},{7}}
 => {{1,2,3,5},{4},{6},{7}}
 => [4,1,1,1] => [4,1,1,1] => ? = 1
{{1,2,3,6,7},{4},{5}}
 => {{1,2,3,7},{4},{5,6}}
 => [4,1,2] => [1,3,1,1,1] => ? = 1
{{1,2,3,6},{4,7},{5}}
 => {{1,2,3},{4,7},{5,6}}
 => [3,2,2] => [1,2,2,1,1] => ? = 2
{{1,2,3,6},{4},{5,7}}
 => {{1,2,3},{4},{5,6,7}}
 => [3,1,3] => [1,1,3,1,1] => ? = 1
{{1,2,3,6},{4},{5},{7}}
 => {{1,2,3},{4},{5,6},{7}}
 => [3,1,2,1] => [2,3,1,1] => ? = 1
{{1,2,3,7},{4,6},{5}}
 => {{1,2,3},{4,6},{5,7}}
 => [3,2,2] => [1,2,2,1,1] => ? = 2
{{1,2,3},{4,6,7},{5}}
 => {{1,2,3,7},{4,6},{5}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3},{4,6},{5,7}}
 => {{1,2,3},{4,6,7},{5}}
 => [3,3,1] => [2,1,2,1,1] => ? = 1
{{1,2,3},{4,6},{5},{7}}
 => {{1,2,3},{4,6},{5},{7}}
 => [3,2,1,1] => [3,2,1,1] => ? = 1
{{1,2,3,7},{4},{5,6}}
 => {{1,2,3,6},{4},{5,7}}
 => [4,1,2] => [1,3,1,1,1] => ? = 1
{{1,2,3},{4,7},{5,6}}
 => {{1,2,3,6},{4,7},{5}}
 => [4,2,1] => [2,2,1,1,1] => ? = 1
{{1,2,3},{4},{5,6,7}}
 => {{1,2,3,6,7},{4},{5}}
 => [5,1,1] => [3,1,1,1,1] => ? = 1
{{1,2,3},{4},{5,6},{7}}
 => {{1,2,3,6},{4},{5},{7}}
 => [4,1,1,1] => [4,1,1,1] => ? = 1
{{1,2,3,7},{4},{5},{6}}
 => {{1,2,3},{4},{5},{6,7}}
 => [3,1,1,2] => [1,4,1,1] => ? = 1
{{1,2,3},{4,7},{5},{6}}
 => {{1,2,3},{4},{5,7},{6}}
 => [3,1,2,1] => [2,3,1,1] => ? = 1
{{1,2,3},{4},{5,7},{6}}
 => {{1,2,3},{4,7},{5},{6}}
 => [3,2,1,1] => [3,2,1,1] => ? = 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001481The minimal height of a peak of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.