Your data matches 44 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001687
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => 0
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,2,3,5,4] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,4,5,2] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,3,2,5,4] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,5,2,3] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => 1
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St001685
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001685: 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] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => [3,2,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [3,1,2] => 0
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => [3,2,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [5,4,3,1,2] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,2,4,3,5] => [5,4,2,3,1] => 0
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => [5,4,1,3,2] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => [5,4,1,3,2] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,4,5,2] => [5,3,2,1,4] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => [5,3,2,4,1] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => [5,3,1,4,2] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,3,2,4,5] => [5,3,4,2,1] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,4,2] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,3,2,5,4] => [5,3,4,1,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,3,2,4,5] => [5,3,4,2,1] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,5,2,3] => [5,2,1,4,3] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => [5,2,4,3,1] => 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000562
Mapping
St000562: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => ? = 0
{{1,2}}
 => 0
{{1},{2}}
 => 0
{{1,2,3}}
 => 0
{{1,2},{3}}
 => 0
{{1,3},{2}}
 => 0
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 0
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 0
{{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 0
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 0
{{1},{2,4},{3}}
 => 0
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 0
{{1,2,3,5},{4}}
 => 0
{{1,2,3},{4,5}}
 => 0
{{1,2,3},{4},{5}}
 => 0
{{1,2,4,5},{3}}
 => 0
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 0
{{1,2},{3,4,5}}
 => 0
{{1,2},{3,4},{5}}
 => 0
{{1,2,5},{3},{4}}
 => 0
{{1,2},{3,5},{4}}
 => 0
{{1,2},{3},{4,5}}
 => 0
{{1,2},{3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => 0
{{1,3,4},{2,5}}
 => 2
{{1,3,4},{2},{5}}
 => 2
{{1,3,5},{2,4}}
 => 1
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 1
{{1,3,5},{2},{4}}
 => 1
{{1,3},{2,5},{4}}
 => 1
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => 0
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 1
Description
The number of internal points of a set partition. An element $e$ is internal, if there are $f < e < g$ such that the blocks of $f$ and $g$ have larger minimal element than the block of $e$. See Section 5.5 of [1]
Matching statistic: St001384
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1]
 => []
 => ?
 => ? = 0
{{1,2}}
 => [2]
 => []
 => ?
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0}
{{1,2,3}}
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0}
{{1,2},{3}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0}
{{1,3},{2}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0}
{{1},{2,3}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0}
{{1},{2},{3}}
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,4}}
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1,2,3},{4}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1,2,4},{3}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1,2},{3,4}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3,4},{2}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1,3},{2,4}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2,3}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
{{1},{2,3},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2},{3}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2,3,4,5}}
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2,3,4},{5}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2,3,5},{4}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2,3},{4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2,4},{3,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2},{3,4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3,4,5},{2}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,3,4},{2,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,3},{2,4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4,5},{2,3}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,4},{2,3,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1},{2,3,4,5}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
{{1,2,3,4,5,6}}
 => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3,4},{5,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3,5},{4,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3},{4,5,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,4,5},{3,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,4,6},{3,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,4},{3,5,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,5,6},{3,4}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,5},{3,4,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2,6},{3,4,5}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,2},{3,4,5,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,3,4,5},{2,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
{{1,3,4,6},{2,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St000934
(load all 2 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%
Values
{{1}}
 => [1]
 => [1]
 => []
 => ? = 0
{{1,2}}
 => [2]
 => [2]
 => []
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0}
{{1,2,3}}
 => [3]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0}
{{1,2},{3}}
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,0}
{{1,3},{2}}
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,0}
{{1},{2,3}}
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,0}
{{1},{2},{3}}
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
{{1,2,3,4}}
 => [4]
 => [2,2]
 => [2]
 => 1
{{1,2,3},{4}}
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
{{1,2,4},{3}}
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
{{1,2},{3,4}}
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
 => [2,1,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
{{1,3},{2,4}}
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
 => [2,1,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,4},{2,3}}
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
{{1},{2,3},{4}}
 => [2,1,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
 => [2,1,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3}}
 => [2,1,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4}}
 => [2,1,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3,4,5}}
 => [5]
 => [2,2,1]
 => [2,1]
 => 0
{{1,2,3,4},{5}}
 => [4,1]
 => [3,2]
 => [2]
 => 1
{{1,2,3,5},{4}}
 => [4,1]
 => [3,2]
 => [2]
 => 1
{{1,2,3},{4,5}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [3,2]
 => [2]
 => 1
{{1,2,4},{3,5}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4,5}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [3,2]
 => [2]
 => 1
{{1,3,4},{2,5}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4,5}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3,5}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,5},{2,3,4}}
 => [3,2]
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4,5}}
 => [4,1]
 => [3,2]
 => [2]
 => 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
{{1,2,3,4,5,6}}
 => [6]
 => [2,2,2]
 => [2,2]
 => 2
{{1,2,3,4,5},{6}}
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [4,2]
 => [2]
 => 1
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [4,1,1]
 => [1,1]
 => 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [4,2]
 => [2]
 => 1
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [4,1,1]
 => [1,1]
 => 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [4,2]
 => [2]
 => 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [4,1,1]
 => [1,1]
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
The 2-degree of an integer partition. For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
Matching statistic: St001604
(load all 3 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Values
{{1}}
 => [1] => [[1],[]]
 => []
 => ? = 0
{{1,2}}
 => [2] => [[2],[]]
 => []
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0}
{{1,2,3}}
 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,0}
{{1,2},{3}}
 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0}
{{1,3},{2}}
 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0}
{{1},{2,3}}
 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4}}
 => [3,1] => [[3,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3}}
 => [3,1] => [[3,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2}}
 => [3,1] => [[3,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4}}
 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3},{4}}
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2},{3}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,4},{3}}
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2},{3,4}}
 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2},{3},{4}}
 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4,5}}
 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,2,3,4},{5}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,2,3},{4},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,2,4},{3},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,2},{3,4,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,2},{3,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,2,5},{3},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,2},{3},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1,3,4,5},{2}}
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,3,4},{2},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,3},{2,4,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,3},{2,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,3,5},{2},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,3},{2},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1,4,5},{2,3}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,4},{2,3,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,4},{2,3},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,5},{2,3,4}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2,3,4,5}}
 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2,3,4},{5}}
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,5},{2,3},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1},{2,3,5},{4}}
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2,3},{4,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,4,5},{2},{3}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1,4},{2},{3,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1,5},{2,4},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
{{1},{2,4,5},{3}}
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2,4},{3,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2,4},{3},{5}}
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,5},{2},{3,4}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2,5},{3,4}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2},{3,4,5}}
 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,5},{2},{3},{4}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2}
{{1,2,3,4,5},{6}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,3,4},{5,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3,4},{5},{6}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,3,5,6},{4}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,3,5},{4,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3,5},{4},{6}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,3,6},{4,5}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3},{4,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,3,6},{4},{5}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
{{1,2,4,5,6},{3}}
 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
{{1,2,4,5},{3,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,4,5},{3},{6}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,4,6},{3,5}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,4},{3,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,4,6},{3},{5}}
 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
{{1,2,4},{3,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2,4},{3},{5,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
{{1,2,5,6},{3,4}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,5},{3,4},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2},{3,4,5},{6}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
{{1,2},{3,4,6},{5}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000771
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 75%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2}}
 => [2] => [1] => ([],1)
 => 1 = 0 + 1
{{1},{2}}
 => [1,1] => [2] => ([],2)
 => ? = 0 + 1
{{1,2,3}}
 => [3] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,1,1] => [3] => ([],3)
 => ? = 0 + 1
{{1,2,3,4}}
 => [4] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,4},{3}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,3,4},{2}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,4},{2,3}}
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1},{2,3,4}}
 => [1,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,2,3,4,5}}
 => [5] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,5},{2,3,4}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3,4,5}}
 => [1,4] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3,4},{5}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,5},{2,3},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,4,5},{2},{3}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,5},{2,4},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2,4,5},{3}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2,4},{3,5}}
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1},{2,4},{3},{5}}
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2,5},{3,4}}
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1},{2},{3,5},{4}}
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
{{1,2,3,4,5,6}}
 => [6] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3,5},{4},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3},{4,5,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3,6},{4},{5}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4,5},{3},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4},{3,5,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4,6},{3},{5}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,5},{3,4,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,6},{3,4,5}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,5,6},{3},{4}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,3,4,5},{2},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,3,4},{2,5,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,3,4,6},{2},{5}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 75%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2}}
 => [2] => [1] => ([],1)
 => 1 = 0 + 1
{{1},{2}}
 => [1,1] => [2] => ([],2)
 => ? = 0 + 1
{{1,2,3}}
 => [3] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2}}
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,1,1] => [3] => ([],3)
 => ? = 0 + 1
{{1,2,3,4}}
 => [4] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,4},{3}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,3,4},{2}}
 => [3,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,4},{2,3}}
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1},{2,3,4}}
 => [1,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0} + 1
{{1,2,3,4,5}}
 => [5] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,5},{2,3,4}}
 => [2,3] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3,4,5}}
 => [1,4] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2,3,4},{5}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,5},{2,3},{4}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,4,5},{2},{3}}
 => [3,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,5},{2,4},{3}}
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2,4,5},{3}}
 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2,4},{3,5}}
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1},{2,4},{3},{5}}
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1},{2,5},{3,4}}
 => [1,2,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1},{2},{3,5},{4}}
 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2} + 1
{{1,2,3,4,5,6}}
 => [6] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2,3,4},{5},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3,5},{4},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3},{4,5,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3,6},{4},{5}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4,5},{3},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4},{3,5,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4,6},{3},{5}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,5},{3,4,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,6},{3,4,5}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,5,6},{3},{4}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,3,4,5},{2},{6}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,3,4},{2,5,6}}
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
{{1,3,4,6},{2},{5}}
 => [4,1,1] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3} + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001876
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [[1],[]]
 => ([],1)
 => ? = 0
{{1,2}}
 => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0}
{{1,2,3}}
 => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1,2},{3}}
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1,3},{2}}
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1},{2,3}}
 => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
 => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4}}
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3}}
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4}}
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2}}
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4}}
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3}}
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4}}
 => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3},{4}}
 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2},{3}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,4},{3}}
 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2},{3,4}}
 => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2},{3},{4}}
 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4,5}}
 => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3,4},{5}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3,5},{4}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4,5}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4,5},{3}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3,5}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,5},{3,4}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4,5}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4,5},{2}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2,5}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,5},{2,4}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4,5}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4,5},{2,3}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3,5}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,5},{2,3,4}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4,5}}
 => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4},{5}}
 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,5},{2,3},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2,3,5},{4}}
 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,4},{2,5},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,5},{2,4},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2,4},{3,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2,5},{3,4}}
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,5,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,4,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,4},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,6},{3,4,5}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,4,5},{6}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2,6},{3,4},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,4,6},{5}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,6},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,5,6},{4}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,2},{3},{4,5},{6}}
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3},{4,6},{5}}
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3,4},{2,5,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,4},{2,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,4},{2,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,6},{2,4,5}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,5},{6}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3,6},{2,4},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,6},{5}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,6},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,5,6},{4}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [[1],[]]
 => ([],1)
 => ? = 0
{{1,2}}
 => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0}
{{1,2,3}}
 => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1,2},{3}}
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1,3},{2}}
 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1},{2,3}}
 => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
 => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4}}
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3}}
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4}}
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2}}
 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4}}
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2},{4}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3}}
 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4}}
 => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3},{4}}
 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2},{3}}
 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,4},{3}}
 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2},{3,4}}
 => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2},{3},{4}}
 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4,5}}
 => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3,4},{5}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3,5},{4}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4,5}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4,5},{3}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3,5}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2,5},{3,4}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4,5}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4,5},{2}}
 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2,5}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2},{5}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3,5},{2,4}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4,5}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2},{4}}
 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,5},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4,5},{2,3}}
 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3,5}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3},{5}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,5},{2,3,4}}
 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4,5}}
 => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4},{5}}
 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1,5},{2,3},{4}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2,3,5},{4}}
 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,4},{2,5},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,5},{2,4},{3}}
 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2,4},{3,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1},{2,5},{3,4}}
 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,5,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,4},{3,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,4,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,4},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,6},{3,4,5}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,4,5},{6}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2,6},{3,4},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,4,6},{5}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,5},{3,6},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,5,6},{4}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,2},{3},{4,5},{6}}
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,2},{3},{4,6},{5}}
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3,4},{2,5,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,4},{2,5},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,4},{2,6},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4,6}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,4},{6}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,6},{2,4,5}}
 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,5},{6}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3,6},{2,4},{5}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,4,6},{5}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,5},{2,6},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
{{1,3},{2,5,6},{4}}
 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
Description
Number of indecomposable injective modules with projective dimension 2.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000455The second largest eigenvalue of a graph if it is integral. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000941The number of characters of the symmetric group whose value on the partition 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. St001720The minimal length of a chain of small intervals in a lattice. St001845The number of join irreducibles minus the rank of a lattice. 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. St000068The number of minimal elements in a poset. St001862The number of crossings 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. St001490The number of connected components of a skew partition. St001867The number of alignments of type EN of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.