- FindStat

Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000007

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000393
(load all 2 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 98%●distinct values known / distinct values provided: 82%

Values

[1,0]
 => [1] => 1 => 1 => 1
[1,0,1,0]
 => [1,1] => 11 => 11 => 2
[1,1,0,0]
 => [2] => 10 => 01 => 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 111 => 3
[1,0,1,1,0,0]
 => [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
 => [2,1] => 101 => 101 => 2
[1,1,0,1,0,0]
 => [3] => 100 => 001 => 2
[1,1,1,0,0,0]
 => [3] => 100 => 001 => 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 1111 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 0111 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 1011 => 3
[1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 0011 => 3
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 0011 => 3
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 1101 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 1001 => 3
[1,1,0,1,0,1,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,0,1,1,0,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 1001 => 3
[1,1,1,0,0,1,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,1,0,1,0,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 0001 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 01111 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 10111 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 11100 => 00111 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 00111 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 11011 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 01011 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 11101 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 01101 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 10101 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 10100 => 00101 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 00101 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 11001 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 10010 => 01001 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 10001 => 10001 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 10001 => 10001 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [5] => 10000 => 00001 => 4
[]
 => [] => ? => ? => ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? => ? => ? => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [11] => 10000000000 => 00000000001 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1,1,1,1,1] => 1011111111 => 1111111101 => ? = 9
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,2,1,1,1,1,1,1] => 1010111111 => 1111110101 => ? = 8
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,2,2,1,1,1,1] => 1010101111 => 1111010101 => ? = 7
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,2,2,1,1] => 1010101011 => 1101010101 => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2,2,2] => 1010101010 => 0101010101 => ? = 5
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => [11] => 10000000000 => 00000000001 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [11] => 10000000000 => 00000000001 => ? = 10
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [11] => 10000000000 => 00000000001 => ? = 10

Description

The number of strictly increasing runs in a binary word.

Matching statistic: St001622

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,5),(4,3),(5,1),(5,2)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,6,8,7] => ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,4,2,7,5,6,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,5,7,6,8] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,2),(5,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,4,6,5,7,8] => ([(0,6),(1,6),(3,7),(4,7),(5,3),(5,4),(6,5),(7,2)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ([(0,7),(1,7),(3,6),(4,6),(5,2),(6,5),(7,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7,8] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,7),(6,7),(7,3)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,5,7,8] => ([(0,6),(0,7),(1,6),(1,7),(3,2),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,3,1,5,6,7,4,8] => ([(0,6),(0,7),(1,3),(2,5),(3,6),(3,7),(4,2),(6,5),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,1,5,7,4,6,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,1,6,4,7,5,8] => ([(0,3),(1,6),(1,7),(2,4),(3,6),(3,7),(5,4),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,1,6,7,4,5,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,4,1,6,8,5,7] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,4,1,7,5,8,6] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,7,8,5,6] => ([(0,6),(0,7),(1,5),(2,6),(2,7),(5,2),(6,4),(7,3)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,8,1,7] => ([(0,7),(1,5),(3,4),(4,6),(5,3),(6,2),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,7,1,8,6] => ([(0,6),(0,7),(1,4),(2,7),(3,5),(4,3),(5,2),(5,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,5,7,8,1,6] => ([(0,7),(1,5),(3,6),(4,2),(5,3),(6,4),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,5,8,1,6,7] => ([(0,7),(1,5),(4,6),(5,4),(6,2),(6,7),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,4,6,1,7,8,5] => ([(0,6),(0,7),(1,4),(2,7),(4,5),(5,2),(5,6),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,4,6,1,8,5,7] => ([(0,6),(0,7),(1,3),(2,5),(2,7),(3,4),(4,2),(4,6),(6,5)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,4,6,7,8,1,5] => ([(0,7),(1,5),(3,4),(4,2),(5,6),(6,3),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,8,1,5,7] => ([(0,7),(1,4),(3,5),(3,7),(4,3),(5,2),(5,6),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,4,7,1,5,8,6] => ([(0,7),(1,5),(3,6),(4,3),(4,7),(5,4),(7,2),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,4,7,1,8,5,6] => ([(0,6),(0,7),(1,5),(3,6),(4,3),(4,7),(5,4),(7,2)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,4,7,8,1,5,6] => ([(0,7),(1,5),(4,2),(5,6),(6,4),(6,7),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,8,1,5,6,7] => ([(0,7),(1,4),(4,5),(5,2),(5,7),(6,3),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,5,1,6,7,8,4] => ([(0,6),(0,7),(1,4),(3,6),(4,3),(4,7),(5,2),(6,5)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [2,3,5,1,6,8,4,7] => ([(0,6),(0,7),(1,4),(3,7),(4,3),(4,6),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,3,5,1,7,8,4,6] => ?
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [2,3,5,1,8,4,6,7] => ([(0,5),(0,6),(1,4),(3,5),(3,7),(4,3),(4,6),(6,7),(7,2)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [2,3,5,6,1,8,4,7] => ([(0,6),(0,7),(1,4),(2,5),(2,7),(3,2),(4,3),(4,6),(6,5)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [2,3,5,6,7,1,8,4] => ([(0,6),(0,7),(1,5),(2,7),(3,4),(4,2),(5,3),(5,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [2,3,5,6,7,8,1,4] => ([(0,7),(1,6),(3,5),(4,3),(5,2),(6,4),(6,7)],8)
 => ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
 => ? = 7 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,3,5,7,1,4,8,6] => ([(0,7),(1,3),(2,6),(3,4),(3,7),(4,2),(4,5),(7,5),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,6,1,4,7,8,5] => ([(0,7),(1,5),(3,6),(5,3),(5,7),(6,4),(7,2),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,3,6,1,7,4,8,5] => ?
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [2,3,6,7,1,4,8,5] => ?
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,6,7,1,8,4,5] => ([(0,6),(0,7),(1,5),(3,6),(4,3),(5,4),(5,7),(7,2)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,3,6,7,8,1,4,5] => ([(0,7),(1,6),(4,5),(5,2),(6,4),(6,7),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,6,8,1,4,5,7] => ?
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,7,1,4,5,8,6] => ([(0,7),(1,4),(3,6),(4,3),(4,7),(5,2),(5,6),(7,5)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,7,1,4,8,5,6] => ([(0,7),(1,5),(3,6),(4,2),(5,3),(5,7),(7,4),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [2,3,7,1,8,4,5,6] => ?
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,8,1,4,5,6,7] => ([(0,7),(1,5),(4,2),(5,3),(5,7),(6,4),(7,6)],8)
 => ?
 => ? = 7 - 1

Description

The number of join-irreducible elements of a lattice. An element

$j$ of a lattice

$L$ is '''join irreducible''' if it is not the least element and if

$j=x\vee y$ , then

$j\in\{x,y\}$ for all

$x,y\in L$ .

Matching statistic: St001615

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 73%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,4,6,7,1,2,5] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [3,5,6,1,7,2,4] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,5,6,7,1,2,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,6,7,1,2,4,5] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,1,6,7,2,3] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,1,2,7,3] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,6,1,7,2,3,5] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,6,7,1,2,3,5] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,1,6,7,2,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,1,2,7,3,4] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,1,7,2,3,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(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)
 => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,5),(4,3),(5,1),(5,2)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,6,8,7] => ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,4,2,7,5,6,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,5,7,6,8] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,2),(5,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,4,6,5,7,8] => ([(0,6),(1,6),(3,7),(4,7),(5,3),(5,4),(6,5),(7,2)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ([(0,7),(1,7),(3,6),(4,6),(5,2),(6,5),(7,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7,8] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,7),(6,7),(7,3)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,5,7,8] => ([(0,6),(0,7),(1,6),(1,7),(3,2),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,3,1,5,6,7,4,8] => ([(0,6),(0,7),(1,3),(2,5),(3,6),(3,7),(4,2),(6,5),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,1,5,7,4,6,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,1,6,4,7,5,8] => ([(0,3),(1,6),(1,7),(2,4),(3,6),(3,7),(5,4),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,1,6,7,4,5,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,4,1,6,8,5,7] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,4,1,7,5,8,6] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,7,8,5,6] => ([(0,6),(0,7),(1,5),(2,6),(2,7),(5,2),(6,4),(7,3)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,8,1,7] => ([(0,7),(1,5),(3,4),(4,6),(5,3),(6,2),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,7,1,8,6] => ([(0,6),(0,7),(1,4),(2,7),(3,5),(4,3),(5,2),(5,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,5,7,8,1,6] => ([(0,7),(1,5),(3,6),(4,2),(5,3),(6,4),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,5,8,1,6,7] => ([(0,7),(1,5),(4,6),(5,4),(6,2),(6,7),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,4,6,1,7,8,5] => ([(0,6),(0,7),(1,4),(2,7),(4,5),(5,2),(5,6),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,4,6,1,8,5,7] => ([(0,6),(0,7),(1,3),(2,5),(2,7),(3,4),(4,2),(4,6),(6,5)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,4,6,7,8,1,5] => ([(0,7),(1,5),(3,4),(4,2),(5,6),(6,3),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,8,1,5,7] => ([(0,7),(1,4),(3,5),(3,7),(4,3),(5,2),(5,6),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,4,7,1,5,8,6] => ([(0,7),(1,5),(3,6),(4,3),(4,7),(5,4),(7,2),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,4,7,1,8,5,6] => ([(0,6),(0,7),(1,5),(3,6),(4,3),(4,7),(5,4),(7,2)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,4,7,8,1,5,6] => ([(0,7),(1,5),(4,2),(5,6),(6,4),(6,7),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,8,1,5,6,7] => ([(0,7),(1,4),(4,5),(5,2),(5,7),(6,3),(7,6)],8)
 => ?
 => ? = 7 - 1

Description

The number of join prime elements of a lattice. An element

$x$ of a lattice

$L$ is join-prime (or coprime) if

$x \leq a \vee b$ implies

$x \leq a$ or

$x \leq b$ for every

$a, b \in L$ .

Matching statistic: St001617

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 73%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,4,6,7,1,2,5] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [3,5,6,1,7,2,4] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,5,6,7,1,2,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,6,7,1,2,4,5] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,1,6,7,2,3] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,1,2,7,3] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,6,1,7,2,3,5] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,6,7,1,2,3,5] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,1,6,7,2,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,1,2,7,3,4] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,1,7,2,3,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(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)
 => ? = 6 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(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)
 => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,4,6,5,8,7] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,5),(4,3),(5,1),(5,2)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,3,5,4,6,8,7] => ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
 => ?
 => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,3,5,6,8,7] => ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,6,8,7] => ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,4,2,7,5,6,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,5,7,6,8] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,2),(5,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,4,6,5,7,8] => ([(0,6),(1,6),(3,7),(4,7),(5,3),(5,4),(6,5),(7,2)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ([(0,7),(1,7),(3,6),(4,6),(5,2),(6,5),(7,3),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7,8] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,7),(6,7),(7,3)],8)
 => ?
 => ? = 6 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,5,7,8] => ([(0,6),(0,7),(1,6),(1,7),(3,2),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,3,1,5,6,7,4,8] => ([(0,6),(0,7),(1,3),(2,5),(3,6),(3,7),(4,2),(6,5),(7,4)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,1,5,7,4,6,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,1,6,4,7,5,8] => ([(0,3),(1,6),(1,7),(2,4),(3,6),(3,7),(5,4),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,1,6,7,4,5,8] => ?
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,4,1,6,8,5,7] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,4,1,7,5,8,6] => ([(0,6),(0,7),(1,4),(3,6),(3,7),(4,3),(6,5),(7,2),(7,5)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,7,8,5,6] => ([(0,6),(0,7),(1,5),(2,6),(2,7),(5,2),(6,4),(7,3)],8)
 => ?
 => ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,8,1,7] => ([(0,7),(1,5),(3,4),(4,6),(5,3),(6,2),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,7,1,8,6] => ([(0,6),(0,7),(1,4),(2,7),(3,5),(4,3),(5,2),(5,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,5,7,8,1,6] => ([(0,7),(1,5),(3,6),(4,2),(5,3),(6,4),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,5,8,1,6,7] => ([(0,7),(1,5),(4,6),(5,4),(6,2),(6,7),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,4,6,1,7,8,5] => ([(0,6),(0,7),(1,4),(2,7),(4,5),(5,2),(5,6),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,4,6,1,8,5,7] => ([(0,6),(0,7),(1,3),(2,5),(2,7),(3,4),(4,2),(4,6),(6,5)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,4,6,7,8,1,5] => ([(0,7),(1,5),(3,4),(4,2),(5,6),(6,3),(6,7)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,8,1,5,7] => ([(0,7),(1,4),(3,5),(3,7),(4,3),(5,2),(5,6),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,4,7,1,5,8,6] => ([(0,7),(1,5),(3,6),(4,3),(4,7),(5,4),(7,2),(7,6)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,4,7,1,8,5,6] => ([(0,6),(0,7),(1,5),(3,6),(4,3),(4,7),(5,4),(7,2)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,4,7,8,1,5,6] => ([(0,7),(1,5),(4,2),(5,6),(6,4),(6,7),(7,3)],8)
 => ?
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,8,1,5,6,7] => ([(0,7),(1,4),(4,5),(5,2),(5,7),(6,3),(7,6)],8)
 => ?
 => ? = 7 - 1

Description

The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice

$L$ is a function

$v:L\mapsto\mathbb R$ satisfying

$v(a\vee b) + v(a\wedge b) = v(a) + v(b).$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.

Matching statistic: St001211
(load all 2 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.

Matching statistic: St001492
(load all 4 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.

Matching statistic: St001068

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 64%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St000053

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St000553

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 55%

Values

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

Description

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

The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001875The number of simple modules with projective dimension at most 1. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001626The number of maximal proper sublattices of a lattice.