Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000063
(load all 2 compositions to match this statistic)
Mapping
St000063: 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]
 => 6
[1,1,1]
 => 4
[4]
 => 5
[3,1]
 => 8
[2,2]
 => 6
[2,1,1]
 => 8
[1,1,1,1]
 => 5
[5]
 => 6
[4,1]
 => 10
[3,2]
 => 12
[3,1,1]
 => 12
[2,2,1]
 => 12
[2,1,1,1]
 => 10
[1,1,1,1,1]
 => 6
[6]
 => 7
[5,1]
 => 12
[4,2]
 => 15
[4,1,1]
 => 15
[3,3]
 => 10
[3,2,1]
 => 24
[3,1,1,1]
 => 15
[2,2,2]
 => 10
[2,2,1,1]
 => 15
[2,1,1,1,1]
 => 12
[1,1,1,1,1,1]
 => 7
[7]
 => 8
[6,1]
 => 14
[5,2]
 => 18
[5,1,1]
 => 18
[4,3]
 => 20
[4,2,1]
 => 30
[4,1,1,1]
 => 20
[3,3,1]
 => 20
[3,2,2]
 => 20
[3,2,1,1]
 => 30
[3,1,1,1,1]
 => 18
[2,2,2,1]
 => 20
[2,2,1,1,1]
 => 18
[2,1,1,1,1,1]
 => 14
[1,1,1,1,1,1,1]
 => 8
[8]
 => 9
[7,1]
 => 16
[6,2]
 => 21
[6,1,1]
 => 21
[5,3]
 => 24
[5,2,1]
 => 36
Description
The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000085
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000085: Ordered trees ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 76%
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]
 => [[],[],[]]
 => 6
[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]
 => [[[],[]],[]]
 => 8
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 6
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 8
[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]
 => [[[[],[]]],[]]
 => 10
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 12
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 12
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 12
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 10
[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]
 => [[[[[],[]]]],[]]
 => 12
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[[[]],[]],[]]
 => 15
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => 15
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => 10
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 24
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 15
[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]
 => [[],[[],[[]]]]
 => 15
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[],[[[[],[]]]]]
 => 12
[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]
 => [[[[[[],[]]]]],[]]
 => 14
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[[[]],[]]],[]]
 => 18
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[[[],[[]]]],[]]
 => 18
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[],[]]
 => 20
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => 30
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 20
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => 20
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => 20
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 30
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[],[[[[]],[]]]]
 => 18
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 20
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[],[[[],[[]]]]]
 => 18
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[],[[[[[],[]]]]]]
 => 14
[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]
 => [[[[[[[],[]]]]]],[]]
 => 16
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[[[]],[]]]],[]]
 => 21
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[[],[[]]]]],[]]
 => 21
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[[[]]],[]],[]]
 => 24
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[[[],[],[]]],[]]
 => 36
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[[[]]]]]]]]]]],[]]
 => ? = 12
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[],[]]]]]]]]],[]]
 => ? = 22
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[]],[]]]]]]],[]]
 => ? = 30
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[],[[]]]]]]]],[]]
 => ? = 30
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[[[]]],[]]]]],[]]
 => ? = 36
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[[],[],[]]]]]],[]]
 => ? = 54
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[],[[[]]]]]]],[]]
 => ? = 36
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [[[[[[[]]]],[]]],[]]
 => ? = 40
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [[[[[[],[]],[]]]],[]]
 => ? = 64
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [[[[[[]],[[]]]]],[]]
 => ? = 48
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [[[[[],[[],[]]]]],[]]
 => ? = 64
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[],[[[[]]]]]],[]]
 => ? = 40
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [[[[[[]]]]],[],[]]
 => ? = 42
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [[[[[],[]]],[]],[]]
 => ? = 70
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [[[[[]],[],[]]],[]]
 => ? = 84
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [[[[],[[]],[]]],[]]
 => ? = 84
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [[[[],[],[[]]]],[]]
 => ? = 84
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [[[],[[[],[]]]],[]]
 => ? = 70
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [[],[[[[[]]]]],[]]
 => ? = 42
[5,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [[[[[],[]]]],[[]]]
 => ? = 42
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [[],[[[[],[]]],[]]]
 => ? = 70
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [[],[[[[[[]]]],[]]]]
 => ? = 40
[4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [[],[[[[]],[],[]]]]
 => ? = 84
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [[],[[[],[[]],[]]]]
 => ? = 84
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [[],[[[[[],[]],[]]]]]
 => ? = 64
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [[],[[[[[[[]]],[]]]]]]
 => ? = 36
[3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [[],[[[],[],[[]]]]]
 => ? = 84
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [[],[[[[[]],[[]]]]]]
 => ? = 48
[3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [[[]],[[[[],[]]]]]
 => ? = 42
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [[],[[],[[[],[]]]]]
 => ? = 70
[3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [[],[[[[],[[],[]]]]]]
 => ? = 64
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [[],[[[[[[],[],[]]]]]]]
 => ? = 54
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [[],[[[[[[[[]],[]]]]]]]]
 => ? = 30
[2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[],[],[[[[[]]]]]]
 => ? = 42
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [[],[[[],[[[[]]]]]]]
 => ? = 40
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [[],[[[[[],[[[]]]]]]]]
 => ? = 36
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [[],[[[[[[[],[[]]]]]]]]]
 => ? = 30
[2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [[],[[[[[[[[[],[]]]]]]]]]]
 => ? = 22
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ?
 => ? = 12
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ?
 => ? = 13
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[[],[]]]]]]]]]],[]]
 => ? = 24
[10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[[]],[]]]]]]]],[]]
 => ? = 33
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[[],[[]]]]]]]]],[]]
 => ? = 33
[9,3]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[[[[]]],[]]]]]],[]]
 => ? = 40
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[[[[[[[],[],[]]]]]]],[]]
 => ? = 60
[9,1,1,1]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[[[[],[[[]]]]]]]],[]]
 => ? = 40
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [[[[[[[[]]]],[]]]],[]]
 => ? = 45
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [[[[[[[],[]],[]]]]],[]]
 => ? = 72
[8,2,2]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
 => [[[[[[[]],[[]]]]]],[]]
 => ? = 54
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[[[[],[[],[]]]]]],[]]
 => ? = 72
Description
The number of linear extensions of the tree. We use Knuth's hook length formula for trees [pg.70, 1]. For an ordered tree $T$ on $n$ vertices, the number of linear extensions is $$ \frac{n!}{\prod_{v\in T}|T_v|}, $$ where $T_v$ is the number of vertices of the subtree rooted at $v$.
Matching statistic: St000110
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 71%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 2
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 3
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 4
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 6
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 8
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 6
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 8
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 10
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 12
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 12
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 12
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 10
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 6
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => 7
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 12
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 15
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 15
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 10
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 24
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 15
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 10
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 15
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 12
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 7
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => 8
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => 14
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 18
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 18
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 20
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 30
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 20
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 20
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 20
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 30
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 18
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 20
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 18
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => 14
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => 8
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ? = 9
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => 16
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => 21
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => 21
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 24
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 36
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 24
[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]
 => [2,3,4,5,6,7,8,9,1] => ? = 9
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ? = 10
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => ? = 18
[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]
 => [3,2,4,5,6,7,8,9,1] => ? = 18
[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]
 => [2,3,4,5,6,7,8,9,10,1] => ? = 10
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => ? = 11
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => ? = 20
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => ? = 27
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ? = 27
[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]
 => [4,2,3,5,6,7,8,9,1] => ? = 27
[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]
 => [3,4,2,5,6,7,8,9,1] => ? = 27
[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]
 => [3,2,4,5,6,7,8,9,10,1] => ? = 20
[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]
 => [2,3,4,5,6,7,8,9,10,11,1] => ? = 11
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => ? = 12
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,1,3,4,5,6,7,8,9,10] => ? = 22
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,1,2,4,5,6,7,8,9] => ? = 30
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,3,1,4,5,6,7,8,9] => ? = 30
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [9,4,1,2,3,5,6,7,8] => ? = 36
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ? = 54
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,4,1,5,6,7,8] => ? = 36
[7,4]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => [8,5,1,2,3,4,6,7] => ? = 40
[7,3,1]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [8,4,2,1,3,5,6,7] => ? = 64
[7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [8,3,4,1,2,5,6,7] => ? = 48
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ? = 64
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,5,1,6,7] => ? = 40
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [7,5,2,1,3,4,6] => ? = 70
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [7,4,3,1,2,5,6] => ? = 84
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [7,4,2,3,1,5,6] => ? = 84
[6,2,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [7,3,4,2,1,5,6] => ? = 84
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,5,1,6] => ? = 70
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => ? = 42
[5,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [6,7,2,1,3,4,5] => ? = 42
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,4,5,7,1] => ? = 70
[5,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,2,3,4,5,7,8,1] => ? = 40
[4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,7,1] => ? = 84
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [5,3,4,2,6,7,1] => ? = 84
[4,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,3,2,4,6,7,8,1] => ? = 64
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ? = 36
[3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,6,7,1] => ? = 84
[3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ? = 48
[3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,1,2] => ? = 42
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,7,1] => ? = 70
[3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,3,5,2,6,7,8,1] => ? = 64
[3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => ? = 54
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ? = 30
[2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 42
[2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ? = 40
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ? = 36
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ? = 30
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000100
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 53%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([],2)
 => 2
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => 3
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 4
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => 6
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 8
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 8
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 10
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => 12
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => 12
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => 12
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => 10
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 6
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 7
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
 => 12
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 15
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 15
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 10
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => 24
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 15
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 10
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 15
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
 => 12
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 7
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 8
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
 => ? = 14
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => 18
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => 18
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 20
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 30
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 20
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => 20
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 20
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 30
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => 18
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 20
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => 18
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => ([(1,6),(2,6),(3,5),(5,4),(6,3)],7)
 => ? = 14
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? = 8
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 9
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => ([(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
 => ? = 16
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => 24
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
 => 36
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => 24
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 15
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 40
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 30
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 40
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => 24
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 30
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 30
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 40
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => ([(1,5),(2,5),(3,5),(5,4)],6)
 => 36
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 21
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => ([(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
 => ? = 16
[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]
 => [2,3,4,5,6,7,8,9,1] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
 => ? = 9
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => ([(1,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],10)
 => ? = 10
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => ([(1,8),(2,8),(4,6),(5,4),(6,3),(7,5),(8,7)],9)
 => ? = 18
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
 => ? = 24
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
 => ? = 24
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
 => ? = 42
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,1,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,7,1] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
 => ? = 42
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,5,6,7,8,1] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
 => ? = 24
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 28
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,1] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
 => ? = 24
[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]
 => [3,2,4,5,6,7,8,9,1] => ([(1,8),(2,8),(4,6),(5,4),(6,3),(7,5),(8,7)],9)
 => ? = 18
[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]
 => [2,3,4,5,6,7,8,9,10,1] => ([(1,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],10)
 => ? = 10
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => ([(1,10),(3,5),(4,3),(5,7),(6,4),(7,9),(8,6),(9,2),(10,8)],11)
 => ? = 11
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => ([(1,9),(2,9),(4,5),(5,7),(6,4),(7,3),(8,6),(9,8)],10)
 => ? = 20
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => ([(1,8),(2,4),(4,8),(5,6),(6,3),(7,5),(8,7)],9)
 => ? = 27
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => ([(1,8),(2,4),(4,8),(5,6),(6,3),(7,5),(8,7)],9)
 => ? = 27
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => ([(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
 => ? = 32
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,1,4,5,6,7] => ([(1,7),(2,7),(3,7),(4,6),(6,5),(7,4)],8)
 => ? = 48
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,4,1,5,6,7] => ([(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
 => ? = 32
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [7,5,1,2,3,4,6] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [7,4,2,1,3,5,6] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [7,3,4,1,2,5,6] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? = 42
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [7,3,2,4,1,5,6] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,1,6] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 21
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [6,2,3,4,5,7,1] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,7,1] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [5,2,3,4,6,7,8,1] => ([(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
 => ? = 32
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,7,1] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? = 42
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,7,1] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
 => ? = 56
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,5,6,7,8,1] => ([(1,7),(2,7),(3,7),(4,6),(6,5),(7,4)],8)
 => ? = 48
[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]
 => [4,2,3,5,6,7,8,9,1] => ([(1,8),(2,4),(4,8),(5,6),(6,3),(7,5),(8,7)],9)
 => ? = 27
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 21
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,7,1] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
 => ? = 35
Description
The number of linear extensions of a poset.