- FindStat

Your data matches 510 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010
(load all 27 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [1,2] => [1,1]
 => 2
[1,1,0,0]
 => [2,1] => [2]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,1]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [3,2]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => [3,1,1]
 => 3

Description

The length of the partition.

Matching statistic: St000093
(load all 20 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => ([],2)
 => 2
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000097
(load all 22 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ 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

[1,0]
 => [1] => ([],1)
 => 1
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [1,2] => ([],2)
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3

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
(load all 22 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ 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

[1,0]
 => [1] => ([],1)
 => 1
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [1,2] => ([],2)
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3

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
(load all 11 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ 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

[1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 2
[1,1,0,0]
 => [1,2] => [1,1]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 3

Description

The largest part of an integer partition.

Matching statistic: St000507
(load all 51 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [[1]]
 => 1
[1,0,1,0]
 => [1,2] => [[1,2]]
 => 2
[1,1,0,0]
 => [2,1] => [[1],[2]]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [[1,2,3]]
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [[1],[2],[3]]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,4],[3]]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,3,4],[2]]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[1,3],[2],[4]]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [[1,4],[2],[3]]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [[1,4],[2],[3]]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,2,3,5],[4]]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [[1,2,4,5],[3]]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,2,4],[3],[5]]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [[1,2,5],[3],[4]]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [[1,2,5],[3],[4]]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,5],[2,4]]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[1,3,4,5],[2]]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,3,4],[2,5]]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[1,3,5],[2],[4]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [[1,3,5],[2],[4]]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [[1,4,5],[2],[3]]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [[1,4,5],[2],[3]]
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [[1,4],[2,5],[3]]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [[1,4,5],[2],[3]]
 => 3

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000676
(load all 79 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St000011
(load all 15 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1]
 => [1,0]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,2] => [1,1]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000013
(load all 6 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [1,0]
 => 1
[1,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 2
[1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 3

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000069
(load all 2 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => [2,1] => ([],2)
 => 2
[1,1,0,0]
 => [2,1] => [1,2] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => ([],3)
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 3
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
 => 3

Description

The number of maximal elements of a poset.

The following 500 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000157The number of descents of a standard tableau. St000378The diagonal inversion number of an integer partition. St001427The number of descents of a signed permutation. St000071The number of maximal chains in a poset. St000203The number of external nodes of a binary tree. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000527The width of the poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001203We associate to 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 Dyck path as follows: St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001616The number of neutral elements in a lattice. St001670The connected partition number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000632The jump number of the poset. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001622The number of join-irreducible elements of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000025The number of initial rises of a Dyck path. St000068The number of minimal elements in a poset. St000167The number of leaves of an ordered tree. St000228The size of a partition. St000273The domination number of a graph. St000308The height of the tree associated to a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000395The sum of the heights of the peaks of a Dyck path. St000528The height of a poset. St000544The cop number of a graph. St000662The staircase size of the code of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000808The number of up steps of the associated bargraph. St000916The packing number of a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001128The exponens consonantiae of a partition. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001462The number of factors of a standard tableaux under concatenation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St000012The area of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000703The number of deficiencies of a permutation. St000778The metric dimension of a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001777The number of weak descents in an integer composition. St000925The number of topologically connected components of a set partition. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000306The bounce count of a Dyck path. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000144The pyramid weight of the Dyck path. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000702The number of weak deficiencies of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000984The number of boxes below precisely one peak. St000996The number of exclusive left-to-right maxima of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001270The bandwidth of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001962The proper pathwidth of a graph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000007The number of saliances of the permutation. St000470The number of runs in a permutation. St000822The Hadwiger number of the graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000141The maximum drop size of a permutation. St000159The number of distinct parts of the integer partition. St000161The sum of the sizes of the right subtrees of a binary tree. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001812The biclique partition number of a graph. St000153The number of adjacent cycles of a permutation. St000332The positive inversions of an alternating sign matrix. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000390The number of runs of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St001280The number of parts of an integer partition that are at least two. St000214The number of adjacencies of a permutation. St000246The number of non-inversions of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000292The number of ascents of a binary word. St000300The number of independent sets of vertices of a graph. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000868The aid statistic in the sense of Shareshian-Wachs. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St000015The number of peaks of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000237The number of small exceedances. St000389The number of runs of ones of odd length in a binary word. St000619The number of cyclic descents of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000742The number of big ascents of a permutation after prepending zero. St000991The number of right-to-left minima of a permutation. 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. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000021The number of descents of a permutation. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000084The number of subtrees. St000087The number of induced subgraphs. St000166The depth minus 1 of an ordered tree. St000201The number of leaf nodes in a binary tree. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000553The number of blocks of a graph. St000636The hull number of a graph. St000717The number of ordinal summands of a poset. St000722The number of different neighbourhoods in a graph. St000740The last entry of a permutation. St000843The decomposition number of a perfect matching. St000906The length of the shortest maximal chain in a poset. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001828The Euler characteristic of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000094The depth of an ordered tree. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000386The number of factors DDU in a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000643The size of the largest orbit of antichains under Panyushev complementation. St000710The number of big deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000864The number of circled entries of the shifted recording tableau of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001153The number of blocks with even minimum in a set partition. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St000806The semiperimeter of the associated bargraph. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000061The number of nodes on the left branch of a binary tree. St000216The absolute length of a permutation. St000485The length of the longest cycle of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule 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

$eA$ . St001651The Frankl number of a lattice. St001674The number of vertices of the largest induced star graph in the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000711The number of big exceedences of a permutation. St000142The number of even parts of a partition. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000236The number of cyclical small weak excedances. St000522The number of 1-protected nodes of a rooted tree. St001726The number of visible inversions of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000146The Andrews-Garvan crank of a partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000761The number of ascents in an integer composition. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000143The largest repeated part of a partition. St000299The number of nonisomorphic vertex-induced subtrees. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001152The number of pairs with even minimum in a perfect matching. St000746The number of pairs with odd minimum in a perfect matching. St001430The number of positive entries in a signed permutation. St000242The number of indices that are not cyclical small weak excedances. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001727The number of invisible inversions of a permutation. St000646The number of big ascents of a permutation. St001668The number of points of the poset minus the width of the poset. St001960The number of descents of a permutation minus one if its first entry is not one. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001866The nesting alignments of a signed permutation. St001896The number of right descents of a signed permutations. St000647The number of big descents of a permutation. St001820The size of the image of the pop stack sorting operator. St001487The number of inner corners of a skew partition. St000884The number of isolated descents of a permutation. 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

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001712The number of natural descents of a standard Young tableau. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001343The dimension of the reduced incidence algebra of a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001877Number of indecomposable injective modules with projective dimension 2. 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

$eA$ . St001488The number of corners of a skew partition. St000670The reversal length of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000352The Elizalde-Pak rank of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000919The number of maximal left branches of a binary tree. St000455The second largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001394The genus of a permutation. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000366The number of double descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001115The number of even descents of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000356The number of occurrences of the pattern 13-2. St000850The number of 1/2-balanced pairs in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St000820The number of compositions obtained by rotating the composition. St001060The distinguishing index of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001737The number of descents of type 2 in a permutation. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000663The number of right floats of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001840The number of descents of a set partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000254The nesting number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000387The matching number of a graph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001220The width of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001393The induced matching number of a graph. St001405The number of bonds in a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000004The major index of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000247The number of singleton blocks of a set partition. St000338The number of pixed points of a permutation. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000640The rank of the largest boolean interval in a poset. St000650The number of 3-rises of a permutation. St000653The last descent of a permutation. St000779The tier of a permutation. St000794The mak of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000918The 2-limited packing number of a graph. St000961The shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001388The number of non-attacking neighbors of a permutation. St001471The magnitude of a Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001862The number of crossings of a signed permutation. St001902The number of potential covers of a poset. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.