Your data matches 122 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000883
Mapping
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 3
[2,1,3,4] => 2
[2,1,4,3] => 4
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 3
[1,3,2,4,5] => 2
[1,3,2,5,4] => 4
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St000363
(load all 3 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St000363: 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)
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => 1
[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)
 => 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 1
[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)
 => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[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)
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 1
[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)
 => 2
[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)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[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)
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1
[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)
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[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)
 => 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
[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)
 => 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
Description
The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of $G$. This is also the number of independent sets of maximal cardinality of $G$.
Matching statistic: St000909
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000909: 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)
 => 1
[3,1,2] => ([(1,2)],3)
 => 1
[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)
 => 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 1
[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)
 => 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 1
[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)
 => 1
[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)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 1
[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)
 => 1
[4,3,1,2] => ([(2,3)],4)
 => 1
[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)
 => 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[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)
 => 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1
[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)
 => 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 1
[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)
 => 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000911
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000911: 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)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => 1
[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)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[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)
 => 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1
[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)
 => 2
[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)
 => 1
[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)
 => 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 1
[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)
 => 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 1
[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)
 => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[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)
 => 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[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)
 => 3
[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)
 => 1
[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)
 => 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 4
Description
The number of maximal antichains of maximal size in a poset.
Matching statistic: St000550
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ 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)
 => ([],1)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[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)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
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
Mp00206: Posets —antichains of maximal size⟶ 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)
 => ([],1)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[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)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
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
Mp00206: Posets —antichains of maximal size⟶ 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)
 => ([],1)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[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)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[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)
 => ([],1)
 => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
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: St000189
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ 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)
 => ([],1)
 => ([],1)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => ([],1)
 => 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => ([],1)
 => 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([],1)
 => ([],1)
 => 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([],1)
 => ([],1)
 => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
Description
The number of elements in the poset.
Matching statistic: St001717
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ 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)
 => ([],1)
 => ([],1)
 => 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => ([],1)
 => 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[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)
 => ([],1)
 => ([],1)
 => 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([],1)
 => ([],1)
 => 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([],1)
 => ([],1)
 => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
Description
The largest size of an interval in a poset.
Matching statistic: St001300
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => 0 = 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)
 => 3 = 4 - 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[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)
 => 3 = 4 - 1
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => 0 = 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)
 => 4 = 5 - 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
The following 112 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000993The multiplicity of the largest part of an integer partition. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000120The number of left tunnels of a Dyck path. St000392The length of the longest run of ones in a binary word. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001462The number of factors of a standard tableaux under concatenation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001570The minimal number of edges to add to make a graph Hamiltonian. St000656The number of cuts of a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001118The acyclic chromatic index of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000770The major index of an integer partition when read from bottom to top. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001432The order dimension of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001060The distinguishing index of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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 $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001378The product of the cohook lengths of the integer partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a 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 $eA$. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St001645The pebbling number of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000007The number of saliances of the permutation. St000259The diameter of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001732The number of peaks visible from the left. St000850The number of 1/2-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset.