- FindStat

Your data matches 417 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000937

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 1
[1,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[6] => ([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 4
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[1,6] => ([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 4
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[2,5] => ([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 3
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
[7] => ([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => 3
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [1,1]
 => 1
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [1,1,1]
 => 2
[8] => ([],8)
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 8

Description

The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation

$S^{(2,2)}$ are

$2$ on the conjugacy classes

$(4)$ and

$(2,2)$ ,

$0$ on the conjugacy classes

$(3,1)$ and

$(1,1,1,1)$ , and

$-1$ on the conjugacy class

$(2,1,1)$ . Therefore, the statistic on the partition

$(2,2)$ is

$2$ .

Matching statistic: St000617
(load all 6 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%

Values

[3] => [1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 1 + 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4 = 2 + 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 2 + 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 1 + 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 1 + 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4 = 2 + 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 1 + 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 1 + 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4 = 2 + 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 1 + 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 4 + 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 5 = 3 + 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 6 = 4 + 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,1,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 5 = 3 + 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,1,2,3] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 4 = 2 + 2
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 3 = 1 + 2
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 2
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 2
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,2,3,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [6,7,8,2,3,4,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,3,1,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,3,4,1,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,1,2,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,1,2,3,4] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 3 = 1 + 2
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => 4 = 2 + 2
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 2

Description

The number of global maxima of a Dyck path.

Matching statistic: St001498
(load all 14 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%

Values

[3] => [1,1,1] => [3] => [1,1,1,0,0,0]
 => ? = 1 + 1
[1,3] => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[4] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,1,3] => [1,1,3] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,4] => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,3] => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[5] => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,1,1,3] => [1,1,4] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,4] => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,2,3] => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,5] => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[2,1,3] => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,4] => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,3] => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[6] => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[1,1,1,1,3] => [1,1,5] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,4] => [1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,2,3] => [1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,5] => [1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[1,2,1,3] => [1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,4] => [1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,3,3] => [1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,6] => [1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
[2,1,1,3] => [1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,4] => [1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,2,3] => [1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,5] => [1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[3,1,3] => [1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,4] => [1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,3] => [1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[7] => [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[1,1,1,5] => [1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,3,3] => [1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,2,3] => [1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,3,1,3] => [1,1,3,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,4,3] => [1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,1,2,3] => [1,1,2,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,2,1,3] => [1,1,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,3,3] => [1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[3,1,1,3] => [1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,3] => [1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,1,3] => [1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,4] => [1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[8] => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8 + 1

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

Matching statistic: St000993
(load all 4 compositions to match this statistic)

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%

Values

[3] => [1,1,1] => [[1,1,1],[]]
 => []
 => ? = 1 + 1
[1,3] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[4] => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? = 2 + 1
[1,1,3] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2 = 1 + 1
[1,4] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3 = 2 + 1
[2,3] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[5] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? = 3 + 1
[1,1,1,3] => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2 = 1 + 1
[1,1,4] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3 = 2 + 1
[1,2,3] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 2 = 1 + 1
[1,5] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 4 = 3 + 1
[2,1,3] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2 = 1 + 1
[2,4] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 3 = 2 + 1
[3,3] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[6] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ? = 4 + 1
[1,1,1,1,3] => [5,1,1] => [[5,5,5],[4,4]]
 => [4,4]
 => 2 = 1 + 1
[1,1,1,4] => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => 3 = 2 + 1
[1,1,2,3] => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => [3,3,2]
 => 2 = 1 + 1
[1,1,5] => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => 4 = 3 + 1
[1,2,1,3] => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [3,3,1]
 => 2 = 1 + 1
[1,2,4] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [2,2,2,1]
 => 3 = 2 + 1
[1,3,3] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [2,2,1,1]
 => 2 = 1 + 1
[1,6] => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => 5 = 4 + 1
[2,1,1,3] => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [3,3]
 => 2 = 1 + 1
[2,1,4] => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [2,2,2]
 => 3 = 2 + 1
[2,2,3] => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 2 = 1 + 1
[2,5] => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => 4 = 3 + 1
[3,1,3] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [2,2]
 => 2 = 1 + 1
[3,4] => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => 3 = 2 + 1
[4,3] => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 2 = 1 + 1
[7] => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => ? = 6 + 1
[1,1,1,5] => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => [3,3,3,3]
 => 4 = 3 + 1
[1,1,3,3] => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
 => [3,3,2,2]
 => 2 = 1 + 1
[1,2,2,3] => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
 => [3,3,2,1]
 => 2 = 1 + 1
[1,3,1,3] => [2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]]
 => [3,3,1,1]
 => 2 = 1 + 1
[1,4,3] => [2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]]
 => [2,2,1,1,1]
 => 2 = 1 + 1
[2,1,2,3] => [1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]]
 => [3,3,2]
 => 2 = 1 + 1
[2,2,1,3] => [1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]]
 => [3,3,1]
 => 2 = 1 + 1
[2,3,3] => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => [2,2,1,1]
 => 2 = 1 + 1
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 1 + 1
[3,2,3] => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => [2,2,1]
 => 2 = 1 + 1
[4,1,3] => [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
 => ?
 => ? = 1 + 1
[4,4] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => [1,1,1]
 => 3 = 2 + 1
[8] => [1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
 => []
 => ? = 8 + 1

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St000287
(load all 19 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000287: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

[3] => ([],3)
 => ([],3)
 => 3 = 1 + 2
[1,3] => ([(2,3)],4)
 => ([],3)
 => 3 = 1 + 2
[4] => ([],4)
 => ([],4)
 => 4 = 2 + 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 3 = 1 + 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 4 = 2 + 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 3 = 1 + 2
[5] => ([],5)
 => ([],5)
 => 5 = 3 + 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 3 = 1 + 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 4 = 2 + 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,5] => ([(4,5)],6)
 => ([],5)
 => 5 = 3 + 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 3 = 1 + 2
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => 4 = 2 + 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[6] => ([],6)
 => ([],6)
 => 6 = 4 + 2
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 3 = 1 + 2
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 4 = 2 + 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 5 = 3 + 2
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => 4 = 2 + 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,6] => ([(5,6)],7)
 => ([],6)
 => 6 = 4 + 2
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => 4 = 2 + 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,5] => ([(4,6),(5,6)],7)
 => ([(4,5)],6)
 => 5 = 3 + 2
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(4,5)],6)
 => 4 = 2 + 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[7] => ([],7)
 => ([],7)
 => ? = 6 + 2
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 2
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 2
[8] => ([],8)
 => ?
 => ? = 8 + 2

Description

The number of connected components of a graph.

Matching statistic: St000553
(load all 8 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

[3] => ([],3)
 => ([],3)
 => 3 = 1 + 2
[1,3] => ([(2,3)],4)
 => ([],3)
 => 3 = 1 + 2
[4] => ([],4)
 => ([],4)
 => 4 = 2 + 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 3 = 1 + 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 4 = 2 + 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 3 = 1 + 2
[5] => ([],5)
 => ([],5)
 => 5 = 3 + 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 3 = 1 + 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 4 = 2 + 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,5] => ([(4,5)],6)
 => ([],5)
 => 5 = 3 + 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 3 = 1 + 2
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => 4 = 2 + 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[6] => ([],6)
 => ([],6)
 => 6 = 4 + 2
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 3 = 1 + 2
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 4 = 2 + 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 5 = 3 + 2
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => 4 = 2 + 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,6] => ([(5,6)],7)
 => ([],6)
 => 6 = 4 + 2
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => 4 = 2 + 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,5] => ([(4,6),(5,6)],7)
 => ([(4,5)],6)
 => 5 = 3 + 2
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(4,5)],6)
 => 4 = 2 + 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[7] => ([],7)
 => ([],7)
 => ? = 6 + 2
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 2
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 2
[8] => ([],8)
 => ?
 => ? = 8 + 2

Description

The number of blocks of a graph. A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.

Matching statistic: St001570
(load all 12 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

[3] => ([],3)
 => ([],3)
 => 3 = 1 + 2
[1,3] => ([(2,3)],4)
 => ([],3)
 => 3 = 1 + 2
[4] => ([],4)
 => ([],4)
 => 4 = 2 + 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 3 = 1 + 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => 4 = 2 + 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => 3 = 1 + 2
[5] => ([],5)
 => ([],5)
 => 5 = 3 + 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => 3 = 1 + 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => 4 = 2 + 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,5] => ([(4,5)],6)
 => ([],5)
 => 5 = 3 + 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => 3 = 1 + 2
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => 4 = 2 + 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[6] => ([],6)
 => ([],6)
 => 6 = 4 + 2
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => 3 = 1 + 2
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => 4 = 2 + 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => 5 = 3 + 2
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => 4 = 2 + 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,6] => ([(5,6)],7)
 => ([],6)
 => 6 = 4 + 2
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => 3 = 1 + 2
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => 4 = 2 + 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,5] => ([(4,6),(5,6)],7)
 => ([(4,5)],6)
 => 5 = 3 + 2
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(4,5)],6)
 => 4 = 2 + 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[7] => ([],7)
 => ([],7)
 => ? = 6 + 2
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 3 + 2
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 2 + 2
[8] => ([],8)
 => ?
 => ? = 8 + 2

Description

The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.

Matching statistic: St001331
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

[3] => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[4] => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,4] => ([(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[5] => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,5] => ([(4,5)],6)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[6] => ([],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,6] => ([(5,6)],7)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,5] => ([(4,6),(5,6)],7)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[7] => ([],7)
 => ([],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 3
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 2
[8] => ([],8)
 => ?
 => ?
 => ? = 8

Description

The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.

Matching statistic: St001336
(load all 4 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

[3] => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[4] => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,4] => ([(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[5] => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,5] => ([(4,5)],6)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[6] => ([],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,6] => ([(5,6)],7)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,5] => ([(4,6),(5,6)],7)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[7] => ([],7)
 => ([],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 3
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 2
[8] => ([],8)
 => ?
 => ?
 => ? = 8

Description

The minimal number of vertices in a graph whose complement is triangle-free.

Matching statistic: St000261
(load all 14 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000261: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

[3] => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[4] => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,4] => ([(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[5] => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,5] => ([(4,5)],6)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[6] => ([],6)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,6] => ([(5,6)],7)
 => ([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,5] => ([(4,6),(5,6)],7)
 => ([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[7] => ([],7)
 => ([],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 + 1
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 3 + 1
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 1 + 1
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 2 + 1
[8] => ([],8)
 => ?
 => ?
 => ? = 8 + 1

Description

The edge connectivity of a graph. This is the minimum number of edges that has to be removed to make the graph disconnected.

The following 407 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000286The number of connected components of the complement of a graph. 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. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001458The rank of the adjacency matrix of a graph. St001963The tree-depth of a graph. St000306The bounce count of a Dyck path. St001357The maximal degree of a regular spanning subgraph of a graph. St001828The Euler characteristic of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St000383The last part of an integer composition. St001812The biclique partition number 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. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000451The length of the longest pattern of the form k 1 2. St000989The number of final rises of a permutation. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000226The convexity of a permutation. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000358The number of occurrences of the pattern 31-2. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001549The number of restricted non-inversions between exceedances. 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. St001726The number of visible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000007The number of saliances of the permutation. St000836The number of descents of distance 2 of a permutation. St001130The number of two successive successions in a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001268The size of the largest ordinal summand in the poset. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000181The number of connected components of the Hasse diagram for the poset. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001717The largest size of an interval in a poset. St000738The first entry in the last row of a standard tableau. St000461The rix statistic of a permutation. St000886The number of permutations with the same antidiagonal sums. St000203The number of external nodes of a binary tree. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001959The product of the heights of the peaks of a Dyck path. St000090The variation of a composition. 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. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . 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

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001481The minimal height of a peak of a Dyck path. St001530The depth of a Dyck path. St000094The depth of an ordered tree. St000258The burning number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000916The packing number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001829The common independence number of a graph. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St000004The major index of a permutation. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000056The decomposition (or block) number of a permutation. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000091The descent variation of a composition. St000120The number of left tunnels of a Dyck path. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000354The number of recoils 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. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000794The mak of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001117The game chromatic index of a graph. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000144The pyramid weight of the Dyck path. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000662The staircase size of the code of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000843The decomposition number of a perfect matching. St000877The depth of the binary word interpreted as a path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001566The length of the longest arithmetic progression in a permutation. St001733The number of weak left to right maxima of a Dyck path. St000025The number of initial rises of a Dyck path. St000528The height of a poset. St000756The sum of the positions of the left to right maxima of a permutation. St000918The 2-limited packing 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: St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001343The dimension of the reduced incidence algebra of a poset. St001809The index of the step at the first peak of maximal height in a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001245The cyclic maximal difference between two consecutive entries of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000480The number of lower covers of a partition in dominance order. St000159The number of distinct parts of the integer partition. St000776The maximal multiplicity of an eigenvalue in a graph. St001949The rigidity index of a graph. St000636The hull number of a graph. St000733The row containing the largest entry of a standard tableau. St000481The number of upper covers of a partition in dominance order. St000308The height of the tree associated to a permutation. St000454The largest eigenvalue of a graph if it is integral. St001270The bandwidth of a graph. St001391The disjunction number of a graph. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001459The number of zero columns in the nullspace of a graph. St001725The harmonious chromatic number of a graph. St000301The number of facets of the stable set polytope of a graph. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001271The competition number of a graph. St001642The Prague dimension of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001345The Hamming dimension of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001871The number of triconnected components of a graph. St000050The depth or height of a binary tree. St000177The number of free tiles in the pattern. St000178Number of free entries. St000365The number of double ascents of a permutation. St000837The number of ascents of distance 2 of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St001589The nesting number of a perfect matching. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001256Number of simple reflexive modules that are 2-stable reflexive. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001516The number of cyclic bonds of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001960The number of descents of a permutation minus one if its first entry is not one. St000317The cycle descent number of a permutation. St000463The number of admissible inversions of a permutation. St000732The number of double deficiencies of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001298The number of repeated entries in the Lehmer code of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001644The dimension of a graph. St001684The reduced word complexity of a permutation. St001715The number of non-records in a permutation. St001727The number of invisible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000316The number of non-left-to-right-maxima of a permutation. St000702The number of weak deficiencies of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000963The 2-shifted major index of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001246The maximal difference between two consecutive entries of a permutation. St001590The crossing number of a perfect matching. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000740The last entry of a permutation. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001902The number of potential covers of a poset. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001651The Frankl number of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001769The reflection length of a signed permutation. St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001881The number of factors of a lattice as a Cartesian product of lattices. St001615The number of join prime elements of a lattice. St001616The number of neutral elements in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001703The villainy of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001754The number of tolerances of a finite lattice. St001846The number of elements which do not have a complement in the lattice. St001861The number of Bruhat lower covers of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000479The Ramsey number of a graph. St001545The second Elser number of a connected graph. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001834The number of non-isomorphic minors of a graph. St001833The number of linear intervals in a lattice. St001620The number of sublattices of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001672The restrained domination number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000071The number of maximal chains in a poset. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. 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. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001863The number of weak excedances of a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.