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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001419
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 10 => 01 => 1
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 10 => 01 => 1
[3,2]
=> 10 => 01 => 01 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 100 => 001 => 1
[2,1,1,1]
=> 0111 => 1110 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 011 => 2
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 1100 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 10 => 01 => 1
[5,2]
=> 10 => 01 => 01 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 10 => 01 => 1
[4,2,1]
=> 001 => 100 => 001 => 1
[4,1,1,1]
=> 0111 => 1110 => 0111 => 3
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 001 => 1
[3,2,1,1]
=> 1011 => 1101 => 0111 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 1000 => 0001 => 1
[2,2,1,1,1]
=> 00111 => 11100 => 00111 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 011111 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 2
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 110 => 011 => 2
[5,3]
=> 11 => 11 => 11 => 2
[5,2,1]
=> 101 => 101 => 011 => 2
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St001330
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 80%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 80%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,2]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[6]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 1 = 0 + 1
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8 = 7 + 1
[8]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[6,2]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[5,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,4]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,2,2]
=> 000 => [4] => ([],4)
=> 1 = 0 + 1
[4,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[3,3,2]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,3,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,2,2,1]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[3,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[2,2,2,2]
=> 0000 => [5] => ([],5)
=> 1 = 0 + 1
[2,2,2,1,1]
=> 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,2,1,1,1,1]
=> 001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[2,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 8 + 1
[9]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[8,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[7,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[7,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[6,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[6,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[5,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[4,3,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[4,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[4,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[3,3,2,1]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,2,2,1,1]
=> 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[3,2,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
[2,2,2,1,1,1]
=> 000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[2,2,1,1,1,1,1]
=> 0011111 => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[2,1,1,1,1,1,1,1]
=> 01111111 => [2,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7 + 1
[1,1,1,1,1,1,1,1,1]
=> 111111111 => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 9 + 1
[8,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[7,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[6,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[6,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[6,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[5,4,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[5,2,2,1]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[4,4,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,3]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,3,2,1]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[4,2,2,1,1]
=> 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[4,2,1,1,1,1]
=> 001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[4,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 + 1
[3,3,2,1,1]
=> 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[3,2,2,2,1]
=> 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000022
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [[1]]
=> [1] => 1
[2]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[3]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4
[5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 5
[6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2
[4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 0
[4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 4
[2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 2
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 4
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 6
[7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 1
[6,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => 1
[5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => 1
[5,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 3
[4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => 1
[4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => 1
[4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 3
[3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => 3
[3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => 1
[3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => 3
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 5
[2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => 1
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => 3
[2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 5
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 7
[8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => 2
[6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => 2
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => 2
[5,2,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => 2
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ? => ? = 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? => ? = 3
[8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? => ? = 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ? = 1
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? => ? = 3
[7,4]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? => ? = 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 3
[7,2,2]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? = 1
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? => ? = 3
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? = 5
[6,5]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[6,3,1,1]
=> [4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ? = 3
[6,2,2,1]
=> [4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? = 1
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ? = 3
[6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? = 5
[5,5,1]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 3
[5,4,2]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8] => ? = 1
[5,4,1,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11] => ? = 3
[5,3,3]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? = 3
[5,3,2,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ? = 3
[5,3,1,1,1]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ? = 5
[5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [8,4,3,2,9,10,11,1,5,6,7] => ? = 1
[5,2,2,1,1]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => ? = 3
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? => ? = 5
[5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? = 7
[4,4,3]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 1
[4,4,2,1]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ? = 1
[4,4,1,1,1]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => ? = 3
[4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ? = 3
[4,3,2,2]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7] => ? = 1
[4,3,2,1,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 3
[4,3,1,1,1,1]
=> [6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ? = 5
[4,2,2,2,1]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => ? = 1
[4,2,2,1,1,1]
=> [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? => ? = 3
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? => ? = 5
[4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? = 7
[3,3,3,2]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 3
[3,3,3,1,1]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ? = 5
[3,3,2,2,1]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? = 3
[3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 5
[3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? = 7
[3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 1
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 3
[3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? = 5
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? => ? = 7
[3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? = 9
Description
The number of fixed points of a permutation.
Matching statistic: St001126
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 80%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> 0
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 0
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 0
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 2
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 0
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 4
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 2
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 6
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 6
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
Description
Number of simple module that are 1-regular in the corresponding Nakayama algebra.
Matching statistic: St000895
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 70%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0]
=> [1,0]
=> [[1]]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [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]]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [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]]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [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
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [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]]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,-1,1],[0,0,0,0,1,0]]
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [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]]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[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],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0]]
=> 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[1,0,0,-1,1,0],[0,0,0,1,0,0]]
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 4
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[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],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 4
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[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
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [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
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[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],[0,0,0,0,0,0,1]]
=> ? = 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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]]
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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]]
=> ? = 8
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0]]
=> ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[1,0,0,0,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? = 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,-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]]
=> ? = 3
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 5
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,-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],[0,0,0,0,0,0,0,1]]
=> ? = 3
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 5
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 5
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 5
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[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],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 7
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[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]]
=> ? = 3
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[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],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 5
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[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],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 7
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0,0],[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],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 9
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0],[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],[0,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0]]
=> ? = 0
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? = 0
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [[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],[0,0,0,0,1,0,0,0,0,0],[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],[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? = 2
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,-1,1],[1,0,0,0,0,-1,1,0],[0,0,0,0,0,1,0,0]]
=> ? = 2
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,-1,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 2
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [[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],[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? = 4
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 0
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,-1,1,0],[1,0,0,0,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 0
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,-1,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 2
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[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],[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? = 4
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,-1,1,0,0],[1,0,0,-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]]
=> ? = 4
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St000696
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000696: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000696: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 2 = 1 + 1
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 1 = 0 + 1
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 3 = 2 + 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [2,1,3] => 2 = 1 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 4 = 3 + 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => 3 = 2 + 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 1 = 0 + 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => 3 = 2 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 5 = 4 + 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 2 = 1 + 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 2 = 1 + 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 2 = 1 + 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 4 = 3 + 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 2 = 1 + 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 4 = 3 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 6 = 5 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 1 = 0 + 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => 3 = 2 + 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [4,3,2,1,6,5] => 1 = 0 + 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [4,3,2,1,5,6] => 3 = 2 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 3 = 2 + 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [3,2,1,5,4,6] => 3 = 2 + 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [3,2,1,4,5,6] => 5 = 4 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 1 = 0 + 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [2,1,4,3,5,6] => 3 = 2 + 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [2,1,3,4,5,6] => 5 = 4 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 7 = 6 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 2 = 1 + 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 2 = 1 + 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 2 = 1 + 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 4 = 3 + 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? = 1 + 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? = 1 + 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 4 = 3 + 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ? = 3 + 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? = 1 + 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? = 3 + 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => 6 = 5 + 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ? = 1 + 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => ? = 3 + 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 8 = 7 + 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 1 = 0 + 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 3 = 2 + 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 1 = 0 + 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 3 = 2 + 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 3 = 2 + 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ? = 2 + 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => 5 = 4 + 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 1 = 0 + 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => ? = 2 + 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 1 = 0 + 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ? = 2 + 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => 5 = 4 + 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? = 2 + 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ? = 4 + 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ? = 2 + 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? = 4 + 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => 7 = 6 + 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 1 = 0 + 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ? = 2 + 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ? = 4 + 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => 7 = 6 + 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 9 = 8 + 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8] => ? = 1 + 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7] => ? = 1 + 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9] => ? = 1 + 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6] => ? = 1 + 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9] => ? = 3 + 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8] => ? = 1 + 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9] => ? = 3 + 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9] => ? = 1 + 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8] => ? = 1 + 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8,9] => ? = 3 + 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,9] => ? = 1 + 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8,9] => ? = 3 + 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => ? = 3 + 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,9] => ? = 3 + 1
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8,9] => ? = 5 + 1
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,9,8] => ? = 1 + 1
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8,9] => ? = 3 + 1
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8,9] => ? = 5 + 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,9] => ? = 1 + 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8,9] => ? = 3 + 1
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8,9] => ? = 5 + 1
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [8,9,10,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,10,9,8] => ? = 2 + 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [10,8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8,10] => ? = 2 + 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7,10] => ? = 2 + 1
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [10,9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9,10] => ? = 2 + 1
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [5,4,3,2,1,10,9,8,7,6] => ? = 2 + 1
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10] => ? = 2 + 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9] => ? = 2 + 1
[5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [10,9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9,10] => ? = 4 + 1
[5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,10] => ? = 2 + 1
[5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [10,9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9,10] => ? = 4 + 1
[4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [10,9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9,10] => ? = 2 + 1
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [8,9,10,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,10,9,8] => ? = 2 + 1
[4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8,10] => ? = 2 + 1
Description
The number of cycles in the breakpoint graph of a permutation.
The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$.
This graph decomposes into alternating cycles, which this statistic counts.
The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by
$$
\frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}),
$$
where $(x)_n=x(x-1)\dots(x-n+1)$.
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