Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000010
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1]
 => 1
{{1,2}}
 => [2,1] => [2,1] => [1,1]
 => 2
{{1},{2}}
 => [1,2] => [1,2] => [2]
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [2,1]
 => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1]
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,1]
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,1]
 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [3]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [3,1]
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [3,1]
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [2,2]
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [3,1]
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [2,2]
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [2,2]
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [3,1]
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [2,1,1]
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [3,1]
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [3,1]
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [3,1]
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [3,1]
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [3,1]
 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [4]
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [4,1]
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1]
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [3,2]
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,2]
 => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [4,1]
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [3,2]
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [3,2]
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [3,2]
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [2,2,1]
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [3,2]
 => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [3,2]
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [3,2]
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [3,2]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [3,2]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [4,1]
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [3,2]
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [3,1,1]
 => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [3,2]
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [2,2,1]
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [3,2]
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [3,2]
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [3,2]
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [3,2]
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [3,2]
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [4,1]
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [2,2,1]
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [3,2]
 => 2
Description
The length of the partition.
Matching statistic: St000062
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 1
{{1,2}}
 => [2,1] => [2,1] => [1,2] => 2
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [2,1,3] => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [3,1,2] => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [1,3,2] => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,3,1] => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [3,2,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [3,2,1,5,4] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [4,5,2,1,3] => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [4,2,1,5,3] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [3,5,2,1,4] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [5,2,1,4,3] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [2,1,5,3,4] => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [4,3,5,1,2] => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [2,1,5,4,3] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [3,5,4,1,2] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [4,3,1,5,2] => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [2,5,3,1,4] => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [5,3,1,4,2] => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [3,1,5,2,4] => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [4,2,5,1,3] => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [5,2,4,1,3] => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [3,1,5,4,2] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [2,5,4,1,3] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [4,5,1,3,2] => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [4,1,5,2,3] => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [3,2,5,1,4] => 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000097
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
Description
The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000147
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1]
 => 1
{{1,2}}
 => [2,1] => [2,1] => [2]
 => 2
{{1},{2}}
 => [1,2] => [1,2] => [1,1]
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [2,1]
 => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1]
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,1]
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,1]
 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,1,1]
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [2,1,1]
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [2,1,1]
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [2,1,1]
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [2,1,1]
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [2,2]
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [2,1,1]
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [3,1]
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [2,1,1]
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [2,1,1]
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [2,1,1]
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [2,1,1,1]
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,1,1,1]
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [2,1,1,1]
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [2,2,1]
 => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [2,1,1,1]
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [2,2,1]
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [2,1,1,1]
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [3,1,1]
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,2,1]
 => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [2,1,1,1]
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [2,2,1]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [2,1,1,1]
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [3,1,1]
 => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [2,1,1,1]
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [3,1,1]
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [2,2,1]
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [2,2,1]
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [2,1,1,1]
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [2,2,1]
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [2,2,1]
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [3,1,1]
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [2,2,1]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000451
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 2
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [3,2,1] => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 2
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [2,3,1] => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 3
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 3
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 3
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 3
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 3
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 3
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000527
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2,1] => [2,1] => ([],2)
 => 2
{{1},{2}}
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(1,2)],3)
 => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => ([(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St001029
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
Description
The size of the core of a graph. The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St001165
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => [1,0]
 => 1
{{1,2}}
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 3
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001261
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 3
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 3
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 3
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 3
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 3
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
Description
The Castelnuovo-Mumford regularity of a graph.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001471The magnitude of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001393The induced matching number of a graph. St001665The number of pure excedances of a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000260The radius of a connected graph. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001624The breadth of a lattice.