- FindStat

Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000108
(load all 2 compositions to match this statistic)

Mapping

St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 2
[2]
 => 3
[1,1]
 => 3
[3]
 => 4
[2,1]
 => 5
[1,1,1]
 => 4
[4]
 => 5
[3,1]
 => 7
[2,2]
 => 6
[2,1,1]
 => 7
[1,1,1,1]
 => 5
[5]
 => 6
[4,1]
 => 9
[3,2]
 => 9
[3,1,1]
 => 10
[2,2,1]
 => 9
[2,1,1,1]
 => 9
[1,1,1,1,1]
 => 6
[6]
 => 7
[5,1]
 => 11
[4,2]
 => 12
[4,1,1]
 => 13
[3,3]
 => 10
[3,2,1]
 => 14
[3,1,1,1]
 => 13
[2,2,2]
 => 10
[2,2,1,1]
 => 12
[2,1,1,1,1]
 => 11
[1,1,1,1,1,1]
 => 7
[7]
 => 8
[6,1]
 => 13
[5,2]
 => 15
[5,1,1]
 => 16
[4,3]
 => 14
[4,2,1]
 => 19
[4,1,1,1]
 => 17
[3,3,1]
 => 16
[3,2,2]
 => 16
[3,2,1,1]
 => 19
[3,1,1,1,1]
 => 16
[2,2,2,1]
 => 14
[2,2,1,1,1]
 => 15
[2,1,1,1,1,1]
 => 13
[1,1,1,1,1,1,1]
 => 8
[8]
 => 9
[7,1]
 => 15
[6,2]
 => 18
[6,1,1]
 => 19
[5,3]
 => 18
[5,2,1]
 => 24

Description

The number of partitions contained in the given partition.

Matching statistic: St000420
(load all 5 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 97%

Values

[1]
 => [1,0,1,0]
 => 2
[2]
 => [1,1,0,0,1,0]
 => 3
[1,1]
 => [1,0,1,1,0,0]
 => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => 4
[2,1]
 => [1,0,1,0,1,0]
 => 5
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 7
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 6
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 7
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 9
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 9
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 9
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 9
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 7
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 11
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 12
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 13
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 10
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 14
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 13
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 10
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 12
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 11
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 7
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 8
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 13
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 15
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 16
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 14
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 19
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 17
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 16
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 16
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 19
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 16
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 14
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 15
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 13
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 8
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 9
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 15
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => 18
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 19
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 18
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 24
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 21
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 15
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 23
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 9
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 10
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 17
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 21
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 22
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 17
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 10
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 11
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 19
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? = 24
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 25
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 26
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 34
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 29
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 25
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 24
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 19
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 11
[]
 => []
 => ? = 1

Description

The number of Dyck paths that are weakly above a Dyck path.

Matching statistic: St000419
(load all 5 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 97%

Values

[1]
 => [1,0,1,0]
 => 1 = 2 - 1
[2]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[2,1]
 => [1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 6 = 7 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 5 = 6 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 6 = 7 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 6 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 8 = 9 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 8 = 9 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 9 = 10 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 8 = 9 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 8 = 9 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5 = 6 - 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 6 = 7 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 10 = 11 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 11 = 12 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 12 = 13 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 9 = 10 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 13 = 14 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 12 = 13 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 9 = 10 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 11 = 12 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 10 = 11 - 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6 = 7 - 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 8 - 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 12 = 13 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 14 = 15 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 15 = 16 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 13 = 14 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 18 = 19 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 16 = 17 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 15 = 16 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 15 = 16 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 18 = 19 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 15 = 16 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 13 = 14 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 14 = 15 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 12 = 13 - 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7 = 8 - 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 9 - 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 15 - 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => 17 = 18 - 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 18 = 19 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 17 = 18 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 23 = 24 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 20 = 21 - 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 14 = 15 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 22 = 23 - 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 9 - 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 10 - 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 17 - 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 21 - 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 22 - 1
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 17 - 1
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 10 - 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 11 - 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 19 - 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? = 24 - 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 25 - 1
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 26 - 1
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 34 - 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 29 - 1
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 25 - 1
[2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 24 - 1
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 19 - 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 11 - 1
[]
 => []
 => ? = 1 - 1

Description

The number of Dyck paths that are weakly above the Dyck path, except for the path itself.

Matching statistic: St001313
(load all 4 compositions to match this statistic)

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 95%

Values

[1]
 => 10 => 01 => 2
[2]
 => 100 => 011 => 3
[1,1]
 => 110 => 001 => 3
[3]
 => 1000 => 0111 => 4
[2,1]
 => 1010 => 0101 => 5
[1,1,1]
 => 1110 => 0001 => 4
[4]
 => 10000 => 01111 => 5
[3,1]
 => 10010 => 01101 => 7
[2,2]
 => 1100 => 0011 => 6
[2,1,1]
 => 10110 => 01001 => 7
[1,1,1,1]
 => 11110 => 00001 => 5
[5]
 => 100000 => 011111 => 6
[4,1]
 => 100010 => 011101 => 9
[3,2]
 => 10100 => 01011 => 9
[3,1,1]
 => 100110 => 011001 => 10
[2,2,1]
 => 11010 => 00101 => 9
[2,1,1,1]
 => 101110 => 010001 => 9
[1,1,1,1,1]
 => 111110 => 000001 => 6
[6]
 => 1000000 => 0111111 => 7
[5,1]
 => 1000010 => 0111101 => 11
[4,2]
 => 100100 => 011011 => 12
[4,1,1]
 => 1000110 => 0111001 => 13
[3,3]
 => 11000 => 00111 => 10
[3,2,1]
 => 101010 => 010101 => 14
[3,1,1,1]
 => 1001110 => 0110001 => 13
[2,2,2]
 => 11100 => 00011 => 10
[2,2,1,1]
 => 110110 => 001001 => 12
[2,1,1,1,1]
 => 1011110 => 0100001 => 11
[1,1,1,1,1,1]
 => 1111110 => 0000001 => 7
[7]
 => 10000000 => 01111111 => 8
[6,1]
 => 10000010 => 01111101 => 13
[5,2]
 => 1000100 => 0111011 => 15
[5,1,1]
 => 10000110 => 01111001 => 16
[4,3]
 => 101000 => 010111 => 14
[4,2,1]
 => 1001010 => 0110101 => 19
[4,1,1,1]
 => 10001110 => 01110001 => 17
[3,3,1]
 => 110010 => 001101 => 16
[3,2,2]
 => 101100 => 010011 => 16
[3,2,1,1]
 => 1010110 => 0101001 => 19
[3,1,1,1,1]
 => 10011110 => 01100001 => 16
[2,2,2,1]
 => 111010 => 000101 => 14
[2,2,1,1,1]
 => 1101110 => 0010001 => 15
[2,1,1,1,1,1]
 => 10111110 => 01000001 => 13
[1,1,1,1,1,1,1]
 => 11111110 => 00000001 => 8
[8]
 => 100000000 => 011111111 => 9
[7,1]
 => 100000010 => 011111101 => 15
[6,2]
 => 10000100 => 01111011 => 18
[6,1,1]
 => 100000110 => 011111001 => 19
[5,3]
 => 1001000 => 0110111 => 18
[5,2,1]
 => 10001010 => 01110101 => 24
[9]
 => 1000000000 => 0111111111 => ? = 10
[8,1]
 => 1000000010 => 0111111101 => ? = 17
[7,1,1]
 => 1000000110 => 0111111001 => ? = 22
[6,1,1,1]
 => 1000001110 => 0111110001 => ? = 25
[5,1,1,1,1]
 => 1000011110 => 0111100001 => ? = 26
[4,1,1,1,1,1]
 => 1000111110 => 0111000001 => ? = 25
[3,1,1,1,1,1,1]
 => 1001111110 => 0110000001 => ? = 22
[2,1,1,1,1,1,1,1]
 => 1011111110 => 0100000001 => ? = 17
[1,1,1,1,1,1,1,1,1]
 => 1111111110 => 0000000001 => ? = 10
[10]
 => 10000000000 => 01111111111 => ? = 11
[9,1]
 => 10000000010 => 01111111101 => ? = 19
[8,2]
 => 1000000100 => 0111111011 => ? = 24
[8,1,1]
 => 10000000110 => 01111111001 => ? = 25
[7,2,1]
 => 1000001010 => 0111110101 => ? = 34
[7,1,1,1]
 => 10000001110 => 01111110001 => ? = 29
[6,2,1,1]
 => 1000010110 => 0111101001 => ? = 40
[6,1,1,1,1]
 => 10000011110 => 01111100001 => ? = 31
[5,2,1,1,1]
 => 1000101110 => 0111010001 => ? = 42
[5,1,1,1,1,1]
 => 10000111110 => 01111000001 => ? = 31
[4,2,1,1,1,1]
 => 1001011110 => 0110100001 => ? = 40
[4,1,1,1,1,1,1]
 => 10001111110 => 01110000001 => ? = 29
[3,2,1,1,1,1,1]
 => 1010111110 => 0101000001 => ? = 34
[3,1,1,1,1,1,1,1]
 => 10011111110 => 01100000001 => ? = 25
[2,2,1,1,1,1,1,1]
 => 1101111110 => 0010000001 => ? = 24
[2,1,1,1,1,1,1,1,1]
 => 10111111110 => 01000000001 => ? = 19
[1,1,1,1,1,1,1,1,1,1]
 => 11111111110 => 00000000001 => ? = 11
[]
 =>  =>  => ? = 1

Description

The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating

$1$ 's as steps

$(1,0)$ and

$0$ 's as steps

$(0,1)$ . Given a binary word

$w$ , this statistic counts the number of lattice paths from the origin to the same endpoint as

$w$ that stay weakly above

$w$ . See [[St001312]] for this statistic on compositions treated as bounce paths.

Matching statistic: St000070
(load all 2 compositions to match this statistic)

Mapping

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 79%

Values

[1]
 => [[1],[]]
 => ([],1)
 => 2
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => 3
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 3
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 4
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 5
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 4
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 10
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 11
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => 12
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 13
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 10
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => 14
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 13
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 10
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => 12
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 11
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[7]
 => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 8
[6,1]
 => [[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => 13
[5,2]
 => [[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => 15
[5,1,1]
 => [[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => 16
[4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => 14
[4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => 19
[4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => 17
[3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => 16
[3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => 16
[3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => 19
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => 16
[2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => 14
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => 15
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => 13
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 8
[8]
 => [[8],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 9
[7,1]
 => [[7,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => 15
[6,2]
 => [[6,2],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => 18
[6,1,1]
 => [[6,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => 19
[5,3]
 => [[5,3],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => 18
[5,2,1]
 => [[5,2,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => 24
[10]
 => [[10],[]]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ? = 11
[9,1]
 => [[9,1],[]]
 => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ? = 19
[8,2]
 => [[8,2],[]]
 => ([(0,2),(0,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6),(8,9)],10)
 => ? = 24
[8,1,1]
 => [[8,1,1],[]]
 => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10)
 => ? = 25
[7,3]
 => [[7,3],[]]
 => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(5,9),(6,1),(7,5),(7,8),(8,9)],10)
 => ? = 26
[7,2,1]
 => [[7,2,1],[]]
 => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(7,9),(8,1),(8,9)],10)
 => ? = 34
[7,1,1,1]
 => [[7,1,1,1],[]]
 => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10)
 => ? = 29
[6,4]
 => [[6,4],[]]
 => ([(0,2),(0,6),(2,7),(3,1),(4,3),(4,8),(5,4),(5,9),(6,5),(6,7),(7,9),(9,8)],10)
 => ? = 25
[6,3,1]
 => [[6,3,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(4,9),(5,2),(6,4),(6,8),(7,1),(7,8),(8,9)],10)
 => ? = 37
[6,2,2]
 => [[6,2,2],[]]
 => ([(0,6),(0,7),(2,9),(3,5),(4,3),(5,1),(6,4),(6,8),(7,2),(7,8),(8,9)],10)
 => ? = 34
[6,2,1,1]
 => [[6,2,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(7,9),(8,4),(8,9)],10)
 => ? = 40
[6,1,1,1,1]
 => [[6,1,1,1,1],[]]
 => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10)
 => ? = 31
[5,5]
 => [[5,5],[]]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 21
[5,4,1]
 => [[5,4,1],[]]
 => ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
 => ? = 34
[5,3,2]
 => [[5,3,2],[]]
 => ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
 => ? = 37
[5,3,1,1]
 => [[5,3,1,1],[]]
 => ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 42
[5,2,2,1]
 => [[5,2,2,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 41
[5,2,1,1,1]
 => [[5,2,1,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(7,9),(8,4),(8,9)],10)
 => ? = 42
[5,1,1,1,1,1]
 => [[5,1,1,1,1,1],[]]
 => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10)
 => ? = 31
[4,4,2]
 => [[4,4,2],[]]
 => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
 => ? = 31
[4,4,1,1]
 => [[4,4,1,1],[]]
 => ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
 => ? = 35
[4,3,3]
 => [[4,3,3],[]]
 => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 30
[4,3,2,1]
 => [[4,3,2,1],[]]
 => ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
 => ? = 42
[4,3,1,1,1]
 => [[4,3,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 41
[4,2,2,2]
 => [[4,2,2,2],[]]
 => ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
 => ? = 35
[4,2,2,1,1]
 => [[4,2,2,1,1],[]]
 => ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
 => ? = 42
[4,2,1,1,1,1]
 => [[4,2,1,1,1,1],[]]
 => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(7,9),(8,4),(8,9)],10)
 => ? = 40
[4,1,1,1,1,1,1]
 => [[4,1,1,1,1,1,1],[]]
 => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10)
 => ? = 29
[3,3,3,1]
 => [[3,3,3,1],[]]
 => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 30
[3,3,2,2]
 => [[3,3,2,2],[]]
 => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
 => ? = 31
[3,3,2,1,1]
 => [[3,3,2,1,1],[]]
 => ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
 => ? = 37
[3,3,1,1,1,1]
 => [[3,3,1,1,1,1],[]]
 => ([(0,6),(0,7),(2,9),(3,5),(4,3),(5,1),(6,4),(6,8),(7,2),(7,8),(8,9)],10)
 => ? = 34
[3,2,2,2,1]
 => [[3,2,2,2,1],[]]
 => ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
 => ? = 34
[3,2,2,1,1,1]
 => [[3,2,2,1,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(4,9),(5,2),(6,4),(6,8),(7,1),(7,8),(8,9)],10)
 => ? = 37
[3,2,1,1,1,1,1]
 => [[3,2,1,1,1,1,1],[]]
 => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(7,9),(8,1),(8,9)],10)
 => ? = 34
[3,1,1,1,1,1,1,1]
 => [[3,1,1,1,1,1,1,1],[]]
 => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10)
 => ? = 25
[2,2,2,2,2]
 => [[2,2,2,2,2],[]]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 21
[2,2,2,2,1,1]
 => [[2,2,2,2,1,1],[]]
 => ([(0,2),(0,6),(2,7),(3,1),(4,3),(4,8),(5,4),(5,9),(6,5),(6,7),(7,9),(9,8)],10)
 => ? = 25
[2,2,2,1,1,1,1]
 => [[2,2,2,1,1,1,1],[]]
 => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(5,9),(6,1),(7,5),(7,8),(8,9)],10)
 => ? = 26
[2,2,1,1,1,1,1,1]
 => [[2,2,1,1,1,1,1,1],[]]
 => ([(0,2),(0,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6),(8,9)],10)
 => ? = 24
[2,1,1,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1,1,1],[]]
 => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ? = 19
[1,1,1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1,1,1],[]]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ? = 11
[]
 => [[],[]]
 => ([],0)
 => ? = 1

Description

The number of antichains in a poset. An antichain in a poset

$P$ is a subset of elements of

$P$ which are pairwise incomparable. An order ideal is a subset

$I$ of

$P$ such that

$a\in I$ and

$b \leq_P a$ implies

$b \in I$ . Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.

Matching statistic: St001664

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001664: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 42%

Values

[1]
 => [[1]]
 => [1] => ([],1)
 => 2
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 3
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 3
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => 5
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => 4
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 5
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 7
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => 7
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => 5
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 9
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 9
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 10
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 9
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 9
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => 6
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 11
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 12
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => 13
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => 10
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => 14
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 13
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => 10
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => 12
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => 11
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => 7
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 8
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 13
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 15
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 16
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 14
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 19
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
 => 17
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 16
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 16
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 19
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => 16
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 14
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => 15
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => 13
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([],7)
 => 8
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 9
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 15
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? = 18
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
 => ? = 19
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
 => ? = 18
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
 => ? = 24
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
 => ? = 21
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ? = 15
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
 => ? = 23
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
 => ? = 22
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
 => ? = 26
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
 => ? = 21
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
 => ? = 19
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
 => ? = 22
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
 => ? = 23
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
 => ? = 24
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
 => ? = 19
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = 15
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
 => ? = 18
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
 => ? = 18
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => ([(6,7)],8)
 => ? = 15
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => ([],8)
 => ? = 9
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 10
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 17
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ([(0,8),(1,3),(4,5),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 21
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ([(2,8),(4,5),(5,7),(6,4),(7,3),(8,6)],9)
 => ? = 22
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
 => ? = 22
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ([(1,8),(2,4),(5,7),(6,5),(7,3),(8,6)],9)
 => ? = 29
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ([(3,8),(5,7),(6,5),(7,4),(8,6)],9)
 => ? = 25
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
 => ? = 20
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
 => ? = 30
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ([(0,5),(1,4),(2,8),(6,7),(7,3),(8,6)],9)
 => ? = 28
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ([(2,8),(3,5),(6,7),(7,4),(8,6)],9)
 => ? = 33
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ([(4,5),(5,8),(7,6),(8,7)],9)
 => ? = 26
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
 => ? = 25
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
 => ? = 28
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
 => ? = 32
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ([(1,6),(2,5),(3,8),(7,4),(8,7)],9)
 => ? = 32
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => ([(3,5),(4,6),(6,7),(7,8)],9)
 => ? = 33
[4,1,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,1,2,3,4] => ([(5,6),(6,8),(8,7)],9)
 => ? = 25

Description

The number of non-isomorphic subposets of a poset.

Matching statistic: St000087

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000087: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 32%

Values

[1]
 => [[1]]
 => [1] => ([],1)
 => 1 = 2 - 1
[2]
 => [[1,2]]
 => [1,2] => ([],2)
 => 2 = 3 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => ([(0,1)],2)
 => 2 = 3 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => 3 = 4 - 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 4 = 5 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 4 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([],4)
 => 4 = 5 - 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 6 = 7 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 5 = 6 - 1
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6 = 7 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([],5)
 => 5 = 6 - 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 8 = 9 - 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 8 = 9 - 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9 = 10 - 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 8 = 9 - 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8 = 9 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([],6)
 => 6 = 7 - 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10 = 11 - 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 11 = 12 - 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 12 = 13 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 9 = 10 - 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 13 = 14 - 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(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)
 => 12 = 13 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 9 = 10 - 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 11 = 12 - 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(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)
 => 10 = 11 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([(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)
 => 6 = 7 - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([],7)
 => ? = 8 - 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 13 - 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 15 - 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 16 - 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 14 - 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 19 - 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 17 - 1
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 16 - 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 16 - 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 19 - 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(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)
 => ? = 16 - 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 14 - 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 15 - 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(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)
 => ? = 13 - 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([(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)
 => ? = 8 - 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([],8)
 => ? = 9 - 1
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 15 - 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 18 - 1
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19 - 1
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ? = 18 - 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 24 - 1
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(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)
 => ? = 21 - 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
 => ? = 15 - 1
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 23 - 1
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 22 - 1
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 26 - 1
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21 - 1
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 19 - 1
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 22 - 1
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 23 - 1
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(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)
 => ? = 24 - 1
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19 - 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 15 - 1
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18 - 1
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18 - 1
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15 - 1
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 9 - 1
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ([],9)
 => ? = 10 - 1
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 17 - 1
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
 => ? = 21 - 1
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 22 - 1
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ? = 22 - 1
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
 => ? = 29 - 1
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 25 - 1
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
 => ? = 20 - 1
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 30 - 1
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
 => ? = 28 - 1
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 33 - 1
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 26 - 1
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 25 - 1

Description

The number of induced subgraphs. A subgraph

$H \subseteq G$ is induced if

$E(H)$ consists of all edges in

$E(G)$ that connect the vertices of

$H$ .

Matching statistic: St001616

Mapping

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 21%

Values

[1]
 => [[1],[]]
 => ([],1)
 => ([(0,1)],2)
 => 2
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 9
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 9
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 10
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 9
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 9
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 11
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 12
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 13
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 10
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? = 14
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 13
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 10
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 12
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 11
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[7]
 => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 8
[6,1]
 => [[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
 => ? = 13
[5,2]
 => [[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => ? = 15
[5,1,1]
 => [[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? = 16
[4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ? = 14
[4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
 => ? = 19
[4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
 => ? = 17
[3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ? = 16
[3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ? = 16
[3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
 => ? = 19
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? = 16
[2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ? = 14
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => ? = 15
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
 => ? = 13
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 8
[8]
 => [[8],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 9
[7,1]
 => [[7,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
 => ? = 15
[6,2]
 => [[6,2],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
 => ? = 18
[6,1,1]
 => [[6,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
 => ? = 19
[5,3]
 => [[5,3],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
 => ? = 18
[5,2,1]
 => [[5,2,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ?
 => ? = 24
[5,1,1,1]
 => [[5,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ?
 => ? = 21
[4,4]
 => [[4,4],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 15
[4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 23
[4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 22
[4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => ?
 => ? = 26
[4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ?
 => ? = 21
[3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ? = 19
[3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 22
[3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 23
[3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ?
 => ? = 24
[3,1,1,1,1,1]
 => [[3,1,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
 => ? = 19
[2,2,2,2]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 15
[2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
 => ? = 18
[2,2,1,1,1,1]
 => [[2,2,1,1,1,1],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
 => ? = 18
[2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
 => ? = 15
[1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 9
[9]
 => [[9],[]]
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ? = 10
[8,1]
 => [[8,1],[]]
 => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
 => ([(0,9),(1,10),(2,11),(3,7),(3,10),(4,6),(4,12),(5,4),(5,14),(6,8),(6,13),(7,5),(7,16),(8,2),(8,15),(9,1),(9,3),(10,16),(12,13),(13,15),(14,12),(15,11),(16,14)],17)
 => ? = 17
[7,2]
 => [[7,2],[]]
 => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9)
 => ?
 => ? = 21
[7,1,1]
 => [[7,1,1],[]]
 => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
 => ?
 => ? = 22
[6,3]
 => [[6,3],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9)
 => ?
 => ? = 22

Description

The number of neutral elements in a lattice. An element

$e$ of the lattice

$L$ is neutral if the sublattice generated by

$e$ ,

$x$ and

$y$ is distributive for all

$x, y \in L$ .

Matching statistic: St001846

Mapping

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 21%

Values

[1]
 => [[1],[]]
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 5 - 2
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 3 = 5 - 2
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 5 = 7 - 2
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4 = 6 - 2
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 5 = 7 - 2
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 3 = 5 - 2
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 4 = 6 - 2
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 7 = 9 - 2
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 7 = 9 - 2
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 10 - 2
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 7 = 9 - 2
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 7 = 9 - 2
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 4 = 6 - 2
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 5 = 7 - 2
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 11 - 2
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 12 - 2
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 13 - 2
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 10 - 2
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? = 14 - 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 13 - 2
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 10 - 2
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 12 - 2
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 11 - 2
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 5 = 7 - 2
[7]
 => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 6 = 8 - 2
[6,1]
 => [[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
 => ? = 13 - 2
[5,2]
 => [[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => ? = 15 - 2
[5,1,1]
 => [[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? = 16 - 2
[4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ? = 14 - 2
[4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
 => ? = 19 - 2
[4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
 => ? = 17 - 2
[3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ? = 16 - 2
[3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ? = 16 - 2
[3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
 => ? = 19 - 2
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? = 16 - 2
[2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ? = 14 - 2
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => ? = 15 - 2
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
 => ? = 13 - 2
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 6 = 8 - 2
[8]
 => [[8],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 9 - 2
[7,1]
 => [[7,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
 => ? = 15 - 2
[6,2]
 => [[6,2],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
 => ? = 18 - 2
[6,1,1]
 => [[6,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
 => ? = 19 - 2
[5,3]
 => [[5,3],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
 => ? = 18 - 2
[5,2,1]
 => [[5,2,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ?
 => ? = 24 - 2
[5,1,1,1]
 => [[5,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ?
 => ? = 21 - 2
[4,4]
 => [[4,4],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 15 - 2
[4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 23 - 2
[4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 22 - 2
[4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => ?
 => ? = 26 - 2
[4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ?
 => ? = 21 - 2
[3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ? = 19 - 2
[3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 22 - 2
[3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 23 - 2
[3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ?
 => ? = 24 - 2
[3,1,1,1,1,1]
 => [[3,1,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
 => ? = 19 - 2
[2,2,2,2]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 15 - 2
[2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
 => ? = 18 - 2
[2,2,1,1,1,1]
 => [[2,2,1,1,1,1],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
 => ? = 18 - 2
[2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
 => ? = 15 - 2
[1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 9 - 2
[9]
 => [[9],[]]
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => ? = 10 - 2
[8,1]
 => [[8,1],[]]
 => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
 => ([(0,9),(1,10),(2,11),(3,7),(3,10),(4,6),(4,12),(5,4),(5,14),(6,8),(6,13),(7,5),(7,16),(8,2),(8,15),(9,1),(9,3),(10,16),(12,13),(13,15),(14,12),(15,11),(16,14)],17)
 => ? = 17 - 2
[7,2]
 => [[7,2],[]]
 => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9)
 => ?
 => ? = 21 - 2
[7,1,1]
 => [[7,1,1],[]]
 => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
 => ?
 => ? = 22 - 2
[6,3]
 => [[6,3],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9)
 => ?
 => ? = 22 - 2

Description

The number of elements which do not have a complement in the lattice. A complement of an element

$x$ in a lattice is an element

$y$ such that the meet of

$x$ and

$y$ is the bottom element and their join is the top element.

Matching statistic: St000071
(load all 2 compositions to match this statistic)

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000071: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%

Values

[1]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,1]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[3]
 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 4
[2,1]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 5
[1,1,1]
 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 4
[4]
 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 5
[3,1]
 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 7
[2,2]
 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 6
[2,1,1]
 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 7
[1,1,1,1]
 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => 5
[5]
 => 100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => 6
[4,1]
 => 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
 => ? = 9
[3,2]
 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 9
[3,1,1]
 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
 => ? = 10
[2,2,1]
 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 9
[2,1,1,1]
 => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
 => ? = 9
[1,1,1,1,1]
 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => 6
[6]
 => 1000000 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
 => ? = 7
[5,1]
 => 1000010 => ([(0,4),(0,5),(1,3),(1,9),(1,17),(2,14),(2,19),(3,2),(3,13),(3,18),(4,15),(4,16),(5,1),(5,15),(5,16),(7,11),(8,7),(9,13),(10,8),(11,6),(12,6),(13,14),(14,12),(15,9),(15,10),(16,10),(16,17),(17,8),(17,18),(18,7),(18,19),(19,11),(19,12)],20)
 => ? = 11
[4,2]
 => 100100 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
 => ? = 12
[4,1,1]
 => 1000110 => ([(0,4),(0,5),(1,13),(1,20),(2,3),(2,14),(2,21),(3,8),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,19),(16,10),(16,19),(17,20),(17,21),(18,13),(18,14),(19,11),(19,12),(20,7),(20,15),(21,15),(21,16)],22)
 => ? = 13
[3,3]
 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => 10
[3,2,1]
 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
 => ? = 14
[3,1,1,1]
 => 1001110 => ([(0,4),(0,5),(1,13),(1,20),(2,3),(2,14),(2,21),(3,8),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,19),(16,10),(16,19),(17,20),(17,21),(18,13),(18,14),(19,11),(19,12),(20,7),(20,15),(21,15),(21,16)],22)
 => ? = 13
[2,2,2]
 => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => 10
[2,2,1,1]
 => 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
 => ? = 12
[2,1,1,1,1]
 => 1011110 => ([(0,4),(0,5),(1,3),(1,9),(1,17),(2,14),(2,19),(3,2),(3,13),(3,18),(4,15),(4,16),(5,1),(5,15),(5,16),(7,11),(8,7),(9,13),(10,8),(11,6),(12,6),(13,14),(14,12),(15,9),(15,10),(16,10),(16,17),(17,8),(17,18),(18,7),(18,19),(19,11),(19,12)],20)
 => ? = 11
[1,1,1,1,1,1]
 => 1111110 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
 => ? = 7
[7]
 => 10000000 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
 => ? = 8
[6,1]
 => 10000010 => ([(0,5),(0,6),(1,4),(1,11),(1,20),(2,16),(2,22),(3,2),(3,17),(3,23),(4,3),(4,15),(4,21),(5,18),(5,19),(6,1),(6,18),(6,19),(8,9),(9,13),(10,8),(11,15),(12,10),(13,7),(14,7),(15,17),(16,14),(17,16),(18,11),(18,12),(19,12),(19,20),(20,10),(20,21),(21,8),(21,23),(22,13),(22,14),(23,9),(23,22)],24)
 => ? = 13
[5,2]
 => 1000100 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
 => ? = 15
[5,1,1]
 => 10000110 => ([(0,5),(0,6),(1,4),(1,16),(1,26),(2,17),(2,25),(3,10),(3,22),(4,3),(4,8),(4,18),(5,1),(5,20),(5,21),(6,2),(6,20),(6,21),(8,10),(9,11),(10,13),(11,12),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,22),(18,23),(19,11),(19,23),(20,25),(20,26),(21,16),(21,17),(22,13),(22,24),(23,12),(23,24),(24,14),(24,15),(25,9),(25,19),(26,18),(26,19)],27)
 => ? = 16
[4,3]
 => 101000 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
 => ? = 14
[4,2,1]
 => 1001010 => ([(0,2),(0,3),(1,5),(1,12),(2,18),(2,19),(3,1),(3,18),(3,19),(5,6),(6,7),(7,10),(8,11),(9,8),(10,4),(11,4),(12,6),(12,14),(13,9),(13,15),(14,7),(14,16),(15,8),(15,16),(16,10),(16,11),(17,9),(17,14),(17,15),(18,5),(18,13),(18,17),(19,12),(19,13),(19,17)],20)
 => ? = 19
[4,1,1,1]
 => 10001110 => ([(0,5),(0,6),(1,4),(1,17),(1,27),(2,3),(2,16),(2,26),(3,8),(3,19),(4,9),(4,20),(5,2),(5,21),(5,22),(6,1),(6,21),(6,22),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,23),(18,24),(19,10),(19,23),(20,11),(20,24),(21,26),(21,27),(22,16),(22,17),(23,12),(23,25),(24,13),(24,25),(25,14),(25,15),(26,18),(26,19),(27,18),(27,20)],28)
 => ? = 17
[3,3,1]
 => 110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => ? = 16
[3,2,2]
 => 101100 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => ? = 16
[3,2,1,1]
 => 1010110 => ([(0,2),(0,3),(1,5),(1,12),(2,18),(2,19),(3,1),(3,18),(3,19),(5,6),(6,7),(7,10),(8,11),(9,8),(10,4),(11,4),(12,6),(12,14),(13,9),(13,15),(14,7),(14,16),(15,8),(15,16),(16,10),(16,11),(17,9),(17,14),(17,15),(18,5),(18,13),(18,17),(19,12),(19,13),(19,17)],20)
 => ? = 19
[3,1,1,1,1]
 => 10011110 => ([(0,5),(0,6),(1,4),(1,16),(1,26),(2,17),(2,25),(3,10),(3,22),(4,3),(4,8),(4,18),(5,1),(5,20),(5,21),(6,2),(6,20),(6,21),(8,10),(9,11),(10,13),(11,12),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,22),(18,23),(19,11),(19,23),(20,25),(20,26),(21,16),(21,17),(22,13),(22,24),(23,12),(23,24),(24,14),(24,15),(25,9),(25,19),(26,18),(26,19)],27)
 => ? = 16
[2,2,2,1]
 => 111010 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
 => ? = 14
[2,2,1,1,1]
 => 1101110 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
 => ? = 15
[2,1,1,1,1,1]
 => 10111110 => ([(0,5),(0,6),(1,4),(1,11),(1,20),(2,16),(2,22),(3,2),(3,17),(3,23),(4,3),(4,15),(4,21),(5,18),(5,19),(6,1),(6,18),(6,19),(8,9),(9,13),(10,8),(11,15),(12,10),(13,7),(14,7),(15,17),(16,14),(17,16),(18,11),(18,12),(19,12),(19,20),(20,10),(20,21),(21,8),(21,23),(22,13),(22,14),(23,9),(23,22)],24)
 => ? = 13
[1,1,1,1,1,1,1]
 => 11111110 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
 => ? = 8
[8]
 => 100000000 => ([(0,2),(0,9),(1,11),(2,10),(3,4),(3,13),(4,6),(4,12),(5,3),(5,15),(6,8),(6,14),(7,5),(7,17),(8,1),(8,16),(9,7),(9,10),(10,17),(12,14),(13,12),(14,16),(15,13),(16,11),(17,15)],18)
 => ? = 9
[7,1]
 => 100000010 => ([(0,6),(0,7),(1,3),(1,11),(1,21),(2,18),(2,25),(3,4),(3,17),(3,24),(4,5),(4,20),(4,27),(5,2),(5,19),(5,26),(6,22),(6,23),(7,1),(7,22),(7,23),(9,12),(10,9),(11,17),(12,13),(13,14),(14,15),(15,8),(16,8),(17,20),(18,16),(19,18),(20,19),(21,9),(21,24),(22,10),(22,11),(23,10),(23,21),(24,12),(24,27),(25,15),(25,16),(26,14),(26,25),(27,13),(27,26)],28)
 => ? = 15
[6,2]
 => 10000100 => ([(0,4),(0,5),(1,3),(1,18),(1,22),(2,14),(2,24),(3,2),(3,13),(3,23),(4,19),(4,20),(5,1),(5,19),(5,20),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,14),(14,7),(15,9),(15,17),(16,11),(16,12),(17,10),(17,16),(18,15),(18,23),(19,21),(19,22),(20,18),(20,21),(21,8),(21,15),(22,8),(22,13),(23,17),(23,24),(24,7),(24,16)],25)
 => ? = 18
[6,1,1]
 => 100000110 => ([(0,6),(0,7),(1,15),(1,25),(2,4),(2,14),(2,24),(3,19),(3,30),(4,5),(4,18),(4,31),(5,3),(5,20),(5,29),(6,2),(6,21),(6,22),(7,1),(7,21),(7,22),(9,17),(10,12),(11,13),(12,11),(13,16),(14,18),(15,10),(16,8),(17,8),(18,20),(19,9),(20,19),(21,14),(21,15),(22,24),(22,25),(23,16),(23,17),(24,28),(24,31),(25,10),(25,28),(26,11),(26,27),(27,13),(27,23),(28,12),(28,26),(29,27),(29,30),(30,9),(30,23),(31,26),(31,29)],32)
 => ? = 19
[5,3]
 => 1001000 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
 => ? = 18
[5,2,1]
 => 10001010 => ([(0,3),(0,4),(1,2),(1,14),(1,22),(2,6),(2,15),(3,23),(3,24),(4,1),(4,23),(4,24),(6,8),(7,9),(8,10),(9,13),(10,12),(11,7),(12,5),(13,5),(14,6),(15,8),(15,19),(16,19),(16,20),(17,11),(17,18),(18,7),(18,20),(19,10),(19,21),(20,9),(20,21),(21,12),(21,13),(22,15),(22,16),(23,17),(23,22),(23,25),(24,14),(24,17),(24,25),(25,11),(25,16),(25,18)],26)
 => ? = 24
[5,1,1,1]
 => 100001110 => ([(0,6),(0,7),(1,4),(1,11),(1,28),(2,5),(2,12),(2,27),(3,18),(3,33),(4,20),(4,32),(5,3),(5,19),(5,31),(6,1),(6,29),(6,30),(7,2),(7,29),(7,30),(9,13),(10,14),(11,20),(12,19),(13,15),(14,16),(15,17),(16,8),(17,8),(18,10),(19,18),(20,9),(21,22),(21,25),(22,23),(22,26),(23,15),(23,24),(24,16),(24,17),(25,13),(25,23),(26,14),(26,24),(27,21),(27,31),(28,21),(28,32),(29,27),(29,28),(30,11),(30,12),(31,22),(31,33),(32,9),(32,25),(33,10),(33,26)],34)
 => ? = 21
[4,4]
 => 110000 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => 15
[4,3,1]
 => 1010010 => ([(0,2),(0,3),(1,12),(1,13),(2,18),(2,19),(3,1),(3,18),(3,19),(5,8),(6,5),(7,10),(8,11),(9,7),(10,4),(11,4),(12,9),(12,15),(13,14),(13,15),(14,8),(14,16),(15,7),(15,16),(16,10),(16,11),(17,5),(17,9),(17,14),(18,6),(18,12),(18,17),(19,6),(19,13),(19,17)],20)
 => ? = 23
[4,2,2]
 => 1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
 => ? = 22
[4,2,1,1]
 => 10010110 => ([(0,3),(0,4),(1,15),(1,25),(2,14),(2,24),(3,2),(3,26),(3,27),(4,1),(4,26),(4,27),(6,8),(7,9),(8,10),(9,11),(10,12),(11,13),(12,5),(13,5),(14,6),(15,7),(16,18),(16,23),(17,19),(17,23),(18,8),(18,21),(19,9),(19,22),(20,12),(20,13),(21,10),(21,20),(22,11),(22,20),(23,21),(23,22),(24,6),(24,18),(25,7),(25,19),(26,16),(26,17),(26,24),(26,25),(27,14),(27,15),(27,16),(27,17)],28)
 => ? = 26
[4,1,1,1,1]
 => 100011110 => ([(0,6),(0,7),(1,4),(1,11),(1,28),(2,5),(2,12),(2,27),(3,18),(3,33),(4,20),(4,32),(5,3),(5,19),(5,31),(6,1),(6,29),(6,30),(7,2),(7,29),(7,30),(9,13),(10,14),(11,20),(12,19),(13,15),(14,16),(15,17),(16,8),(17,8),(18,10),(19,18),(20,9),(21,22),(21,25),(22,23),(22,26),(23,15),(23,24),(24,16),(24,17),(25,13),(25,23),(26,14),(26,24),(27,21),(27,31),(28,21),(28,32),(29,27),(29,28),(30,11),(30,12),(31,22),(31,33),(32,9),(32,25),(33,10),(33,26)],34)
 => ? = 21
[3,3,2]
 => 110100 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
 => ? = 19
[3,3,1,1]
 => 1100110 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
 => ? = 22
[3,2,2,1]
 => 1011010 => ([(0,2),(0,3),(1,12),(1,13),(2,18),(2,19),(3,1),(3,18),(3,19),(5,8),(6,5),(7,10),(8,11),(9,7),(10,4),(11,4),(12,9),(12,15),(13,14),(13,15),(14,8),(14,16),(15,7),(15,16),(16,10),(16,11),(17,5),(17,9),(17,14),(18,6),(18,12),(18,17),(19,6),(19,13),(19,17)],20)
 => ? = 23
[3,2,1,1,1]
 => 10101110 => ([(0,3),(0,4),(1,2),(1,14),(1,22),(2,6),(2,15),(3,23),(3,24),(4,1),(4,23),(4,24),(6,8),(7,9),(8,10),(9,13),(10,12),(11,7),(12,5),(13,5),(14,6),(15,8),(15,19),(16,19),(16,20),(17,11),(17,18),(18,7),(18,20),(19,10),(19,21),(20,9),(20,21),(21,12),(21,13),(22,15),(22,16),(23,17),(23,22),(23,25),(24,14),(24,17),(24,25),(25,11),(25,16),(25,18)],26)
 => ? = 24
[3,1,1,1,1,1]
 => 100111110 => ([(0,6),(0,7),(1,15),(1,25),(2,4),(2,14),(2,24),(3,19),(3,30),(4,5),(4,18),(4,31),(5,3),(5,20),(5,29),(6,2),(6,21),(6,22),(7,1),(7,21),(7,22),(9,17),(10,12),(11,13),(12,11),(13,16),(14,18),(15,10),(16,8),(17,8),(18,20),(19,9),(20,19),(21,14),(21,15),(22,24),(22,25),(23,16),(23,17),(24,28),(24,31),(25,10),(25,28),(26,11),(26,27),(27,13),(27,23),(28,12),(28,26),(29,27),(29,30),(30,9),(30,23),(31,26),(31,29)],32)
 => ? = 19
[2,2,2,2]
 => 111100 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => 15
[2,2,2,1,1]
 => 1110110 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
 => ? = 18
[2,2,1,1,1,1]
 => 11011110 => ([(0,4),(0,5),(1,3),(1,18),(1,22),(2,14),(2,24),(3,2),(3,13),(3,23),(4,19),(4,20),(5,1),(5,19),(5,20),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,14),(14,7),(15,9),(15,17),(16,11),(16,12),(17,10),(17,16),(18,15),(18,23),(19,21),(19,22),(20,18),(20,21),(21,8),(21,15),(22,8),(22,13),(23,17),(23,24),(24,7),(24,16)],25)
 => ? = 18
[3,3,3]
 => 111000 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => 20

Description

The number of maximal chains in a poset.

The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice.