Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000063
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 2 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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 5 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: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 91%
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] => ? ∊ {9,9}
[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}
[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] => ? ∊ {10,10,18,18}
[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] => ? ∊ {10,10,18,18}
[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] => ? ∊ {10,10,18,18}
[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] => ? ∊ {10,10,18,18}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
[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] => ? ∊ {11,11,20,20,27,27,27,27}
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: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 62%
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)
 => ? ∊ {7,7}
[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}
[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)
 => ? ∊ {8,8,14,14}
[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)
 => ? ∊ {8,8,14,14}
[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)
 => ? ∊ {8,8,14,14}
[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)
 => ? ∊ {8,8,14,14}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {10,10,18,18,24,24,24,24,28,28,28,28,42,42}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
[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)
 => ? ∊ {11,11,20,20,21,21,27,27,27,27,32,32,32,32,35,35,35,35,42,42,48,48,56,56,56,56}
Description
The number of linear extensions of a poset.
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: 25%
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)
 => ? = 6
[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)
 => ? ∊ {8,8}
[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)
 => ? ∊ {8,8}
[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)
 => ? ∊ {10,10,12,12,12}
[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)
 => ? ∊ {10,10,12,12,12}
[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)
 => ? ∊ {10,10,12,12,12}
[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)
 => ? ∊ {10,10,12,12,12}
[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)
 => ? ∊ {10,10,12,12,12}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {7,7,12,12,15,15,15,15,24}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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)
 => ? ∊ {9,9,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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.
Matching statistic: St001855
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001855: Signed permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 2
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 3
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 4
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 6
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? ∊ {5,5}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [4,2,1,3] => 8
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 6
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 8
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {5,5}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? ∊ {6,6,10,10}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [5,2,1,3,4] => ? ∊ {6,6,10,10}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [4,3,1,2] => 12
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,2,3,1] => 12
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => 12
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => ? ∊ {6,6,10,10}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? ∊ {6,6,10,10}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [6,2,1,3,4,5] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,3,1,2,4] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [5,2,3,1,4] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [4,5,1,2,3] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 24
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [4,2,3,5,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,5,1,2] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,2,4,5,6,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[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] => [2,3,4,5,6,7,1] => ? ∊ {7,7,10,10,12,12,15,15,15,15}
[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] => [8,1,2,3,4,5,6,7] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [7,2,1,3,4,5,6] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [6,3,1,2,4,5] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [6,2,3,1,4,5] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [5,4,1,2,3] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [5,3,2,1,4] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,2,3,4,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,5,2,1,3] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,5,1,2] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [4,2,3,5,6,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,4,2,5,6,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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] => [3,2,4,5,6,7,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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] => [2,3,4,5,6,7,8,1] => ? ∊ {8,8,14,14,18,18,18,18,20,20,20,20,20,30,30}
[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] => [9,1,2,3,4,5,6,7,8] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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] => [8,2,1,3,4,5,6,7] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [7,3,1,2,4,5,6] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [7,2,3,1,4,5,6] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [6,4,1,2,3,5] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [6,3,2,1,4,5] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [6,2,3,4,1,5] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [5,4,2,1,3] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [5,3,4,1,2] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,3,2,4,1] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [5,2,3,4,6,1] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,5,3,1,2] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,5,2,3,1] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [4,3,2,5,6,1] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[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] => [4,2,3,5,6,7,1] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => [3,4,5,2,6,1] => ? ∊ {9,9,15,15,16,16,21,21,21,21,24,24,24,24,30,30,30,36,36,40,40,40}
Description
The number of signed permutations less than or equal to a signed permutation in left weak order.