- FindStat

Your data matches 92 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000011
(load all 125 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000010
(load all 3 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000025
(load all 16 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000147

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000378

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$ . See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].

Matching statistic: St000383
(load all 15 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

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

Description

The last part of an integer composition.

Matching statistic: St001462
(load all 23 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

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

Description

The number of factors of a standard tableaux under concatenation. The concatenation of two standard Young tableaux

$T_1$ and

$T_2$ is obtained by adding the largest entry of

$T_1$ to each entry of

$T_2$ , and then appending the rows of the result to

$T_1$ , see [1, dfn 2.10]. This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.

Matching statistic: St000439
(load all 16 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

[1] => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5 = 4 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5 = 4 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[] => []
 => []
 => ? = 0 + 1

Description

The position of the first down step of a Dyck path.

Matching statistic: St000288
(load all 50 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

[1] => ([],1)
 => [1]
 => 10 => 1
[1,2] => ([],2)
 => [1,1]
 => 110 => 2
[2,1] => ([(0,1)],2)
 => [2]
 => 100 => 1
[1,2,3] => ([],3)
 => [1,1,1]
 => 1110 => 3
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => 1010 => 2
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => 1010 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1000 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => 1000 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 1
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 11110 => 4
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => 10110 => 3
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => 10110 => 3
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 10110 => 3
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 10000 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 10000 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 10000 => 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => 111110 => 5
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 4
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 100010 => 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 100010 => 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 100010 => 2
[] => ([],0)
 => []
 =>  => ? = 0

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000382
(load all 3 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

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

Description

The first part of an integer composition.

The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000733The row containing the largest entry of a standard tableau. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St000234The number of global ascents of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000502The number of successions of a set partitions. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001363The Euler characteristic of a graph according to Knill. St000717The number of ordinal summands of a poset. St000069The number of maximal elements of a poset. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St000286The number of connected components of the complement of a graph. St000153The number of adjacent cycles of a permutation. St000007The number of saliances of the permutation. St000068The number of minimal elements in a poset. St000553The number of blocks of a graph. St000906The length of the shortest maximal chain in a poset. St000546The number of global descents of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St000237The number of small exceedances. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000054The first entry of the permutation. St000989The number of final rises of a permutation. St000456The monochromatic index of a connected graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000843The decomposition number of a perfect matching. St000990The first ascent of a permutation. St000654The first descent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000740The last entry of a permutation. 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. St000308The height of the tree associated to a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 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. St000062The length of the longest increasing subsequence of the permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama 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. St000883The number of longest increasing subsequences of a permutation. St001185The number of indecomposable injective modules of grade at least 2 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. St000061The number of nodes on the left branch of a binary tree. St000924The number of topologically connected components of a perfect matching. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000732The number of double deficiencies of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001570The minimal number of edges to add to make a graph Hamiltonian. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001889The size of the connectivity set of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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.