Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St000071
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => 1
[2,1] => ([],2)
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(1,2)],3)
 => 2
[3,1,2] => ([(1,2)],3)
 => 2
[3,2,1] => ([],3)
 => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => 3
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 3
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(2,3)],4)
 => 3
[4,3,1,2] => ([(2,3)],4)
 => 3
[4,3,2,1] => ([],4)
 => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
Description
The number of maximal chains in a poset.
Matching statistic: St000912
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(1,2)],3)
 => 2
[3,1,2] => ([(1,2)],3)
 => 2
[3,2,1] => ([],3)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[3,4,2,1] => ([(2,3)],4)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => 2
[4,3,2,1] => ([],4)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 4
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 5
Description
The number of maximal antichains in a poset.
Matching statistic: St001304
(load all 4 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St001304: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 4
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 4
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
Description
The number of maximally independent sets of vertices of a graph. An '''independent set''' of vertices of a graph is a set of vertices no two of which are adjacent. If a set of vertices is independent then so is every subset. This statistic counts the number of maximally independent sets of vertices.
Matching statistic: St000010
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00251: Graphs —clique sizes⟶ 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
[1,2] => ([],2)
 => [1,1]
 => 2
[2,1] => ([(0,1)],2)
 => [2]
 => 1
[1,2,3] => ([],3)
 => [1,1,1]
 => 3
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => 2
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 4
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => 3
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => 3
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 3
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => 5
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 4
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 4
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 4
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 4
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 5
Description
The length of the partition.
Matching statistic: St000550
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
Description
The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Matching statistic: St000551
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
Description
The number of left modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Matching statistic: St001616
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
Description
The number of neutral elements in a lattice. An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St000147
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ 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)
 => [1,1]
 => [2]
 => 2
[2,1] => ([(0,1)],2)
 => [2]
 => [1,1]
 => 1
[1,2,3] => ([],3)
 => [1,1,1]
 => [3]
 => 3
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [2,2]
 => [2,2]
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [2,2]
 => [2,2]
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [4]
 => 4
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 3
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 3
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => [3,2]
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => [3,2]
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 3
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => [3,2]
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2,2,1]
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => [3,2]
 => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2,2,1]
 => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [4,4]
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2,2,1]
 => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2,2,1]
 => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [5]
 => 5
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 4
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,2]
 => 4
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,2]
 => 4
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,2]
 => 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [3,2,1]
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,2]
 => 4
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [3,2,1]
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => [5,4]
 => 5
Description
The largest part of an integer partition.
Matching statistic: St000189
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
Description
The number of elements in the poset.
Matching statistic: St001717
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
Description
The largest size of an interval in a poset.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000656The number of cuts of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St001060The distinguishing index of a graph. St000259The diameter of a connected graph. St001769The reflection length of a signed permutation. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000438The position of the last up step in a Dyck path. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St000381The largest part of an integer composition. St001875The number of simple modules with projective dimension at most 1. St001862The number of crossings of a signed permutation. St001896The number of right descents of a signed permutations. St001864The number of excedances of a signed permutation. St001330The hat guessing number of a graph. St001820The size of the image of the pop stack sorting operator. 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 $eA$. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001645The pebbling number of a connected graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. 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. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000456The monochromatic index of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St000632The jump number of the poset. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001877Number of indecomposable injective modules with projective dimension 2.