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Your data matches 137 different statistics following compositions of up to 3 maps.
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Matching statistic: St001722
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St001722: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 1
01 => 1
10 => 1
11 => 1
000 => 1
001 => 1
010 => 1
011 => 1
100 => 1
101 => 1
110 => 1
111 => 1
0000 => 1
0001 => 1
0010 => 1
0011 => 1
0100 => 2
0101 => 1
0110 => 2
0111 => 1
1000 => 1
1001 => 1
1010 => 1
1011 => 1
1100 => 1
1101 => 1
1110 => 1
1111 => 1
00000 => 1
00001 => 1
00010 => 1
00011 => 1
00100 => 3
00101 => 1
00110 => 2
00111 => 1
01000 => 3
01001 => 2
01010 => 1
01011 => 1
01100 => 1
01101 => 3
01110 => 3
01111 => 1
10000 => 1
10001 => 1
10010 => 1
10011 => 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000068
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 27%●distinct values known / distinct values provided: 17%
St000068: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 27%●distinct values known / distinct values provided: 17%
Values
0 => ([(0,1)],2)
=> 1
1 => ([(0,1)],2)
=> 1
00 => ([(0,2),(2,1)],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
11 => ([(0,2),(2,1)],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => ([(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)
=> 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1
00011 => ([(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)
=> 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2
00111 => ([(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)
=> 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 3
01111 => ([(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)
=> 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 2
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 4
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 2
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 6
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 3
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 1
001011 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> ? = 1
001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1
001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 3
001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 3
001111 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 1
010000 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 4
010001 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 3
010010 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2
010011 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 2
010100 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 5
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The number of minimal elements in a poset.
Matching statistic: St001621
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
00011 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 2
00111 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 3
01111 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,9),(2,16),(3,15),(4,12),(5,14),(6,7),(6,15),(7,5),(7,11),(8,1),(9,10),(10,3),(10,6),(11,14),(11,16),(12,8),(13,12),(14,13),(15,2),(15,11),(16,4),(16,13)],17)
=> ? = 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ?
=> ? = 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ?
=> ? = 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 4
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ?
=> ? = 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ?
=> ? = 2
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ?
=> ? = 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 6
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ([(0,11),(1,15),(2,15),(3,15),(4,14),(5,14),(6,14),(7,13),(8,13),(10,12),(11,10),(12,1),(12,2),(12,3),(13,9),(14,7),(14,8),(15,4),(15,5),(15,6)],16)
=> ? = 3
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ([(0,9),(1,16),(3,15),(4,14),(5,13),(6,10),(7,8),(7,14),(8,5),(8,11),(9,6),(10,4),(10,7),(11,13),(11,15),(12,16),(13,12),(14,3),(14,11),(15,1),(15,12),(16,2)],17)
=> ? = 1
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
00011 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 2
00111 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 3
01111 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,9),(2,16),(3,15),(4,12),(5,14),(6,7),(6,15),(7,5),(7,11),(8,1),(9,10),(10,3),(10,6),(11,14),(11,16),(12,8),(13,12),(14,13),(15,2),(15,11),(16,4),(16,13)],17)
=> ? = 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ?
=> ? = 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ?
=> ? = 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 4
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ?
=> ? = 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ?
=> ? = 2
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ?
=> ? = 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 6
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ([(0,11),(1,15),(2,15),(3,15),(4,14),(5,14),(6,14),(7,13),(8,13),(10,12),(11,10),(12,1),(12,2),(12,3),(13,9),(14,7),(14,8),(15,4),(15,5),(15,6)],16)
=> ? = 3
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ([(0,9),(1,16),(3,15),(4,14),(5,13),(6,10),(7,8),(7,14),(8,5),(8,11),(9,6),(10,4),(10,7),(11,13),(11,15),(12,16),(13,12),(14,3),(14,11),(15,1),(15,12),(16,2)],17)
=> ? = 1
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St000093
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 2 = 1 + 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,8),(4,5),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ([(0,4),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,7),(6,7),(7,8)],9)
=> ? = 2 + 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 2 = 1 + 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 2 = 1 + 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ([(0,4),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,7),(6,7),(7,8)],9)
=> ? = 1 + 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,8),(4,5),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 2 = 1 + 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
00001 => ([(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)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
00011 => ([(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)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,4),(2,11),(3,6),(3,9),(4,5),(4,8),(5,10),(6,10),(7,8),(7,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(0,9),(0,11),(1,2),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,9),(6,10),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3 + 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 2 + 1
00111 => ([(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)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3 + 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? = 2 + 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(0,1),(0,4),(0,5),(0,8),(0,9),(1,4),(1,5),(1,8),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(8,9)],10)
=> ? = 1 + 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1 + 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? = 3 + 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,9),(0,11),(1,7),(1,8),(2,5),(2,9),(2,11),(3,4),(3,9),(3,11),(4,7),(4,10),(5,8),(5,10),(6,7),(6,8),(6,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(0,7),(0,8),(0,9),(0,11),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(10,11)],12)
=> ? = 3 + 1
01111 => ([(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)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,9),(0,11),(1,7),(1,8),(2,5),(2,9),(2,11),(3,4),(3,9),(3,11),(4,7),(4,10),(5,8),(5,10),(6,7),(6,8),(6,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(0,7),(0,8),(0,9),(0,11),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(10,11)],12)
=> ? = 1 + 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? = 1 + 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1 + 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? = 2 + 1
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(0,1),(0,4),(0,5),(0,8),(0,9),(1,4),(1,5),(1,8),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(8,9)],10)
=> ? = 1 + 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,8),(2,10),(2,11),(3,6),(3,7),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(5,11),(6,7),(6,9),(6,10),(6,11),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(10,11)],12)
=> ? = 2 + 1
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1 + 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,4),(2,11),(3,6),(3,9),(4,5),(4,8),(5,10),(6,10),(7,8),(7,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(0,9),(0,11),(1,2),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,9),(6,10),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,11),(5,6),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,7),(1,8),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(5,10),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1 + 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,3),(0,11),(1,2),(1,8),(2,9),(3,10),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(6,10),(7,11),(8,9),(10,11)],12)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,10),(1,11),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(3,4),(3,5),(3,7),(3,8),(3,9),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1 + 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,9),(0,14),(1,7),(1,10),(2,8),(2,10),(3,4),(3,9),(3,14),(4,5),(4,11),(5,7),(5,13),(6,8),(6,13),(6,14),(7,12),(8,12),(9,11),(10,12),(11,13),(11,14),(12,13)],15)
=> ([(0,3),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(2,10),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,13),(5,14),(6,8),(6,9),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 1 + 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(0,7),(0,11),(1,6),(1,10),(2,8),(2,10),(3,9),(3,11),(4,8),(4,9),(4,14),(5,6),(5,7),(5,14),(6,12),(7,13),(8,12),(9,13),(10,12),(11,13),(12,14),(13,14)],15)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,14),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 1 + 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(0,7),(0,15),(1,5),(1,14),(2,6),(2,8),(3,4),(3,13),(3,15),(4,6),(4,10),(5,12),(5,13),(6,11),(7,8),(7,9),(8,11),(9,11),(9,12),(9,15),(10,11),(10,12),(10,13),(12,14),(13,14),(14,15)],16)
=> ([(0,1),(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(0,13),(0,14),(0,15),(1,4),(1,5),(1,6),(1,7),(1,10),(1,11),(1,12),(1,13),(1,14),(1,15),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(3,6),(3,7),(3,8),(3,9),(3,10),(3,12),(3,13),(3,14),(3,15),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,13),(4,14),(4,15),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(9,10),(9,12),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,15),(12,13),(12,14),(13,14),(13,15),(14,15)],16)
=> ? = 4 + 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(0,5),(0,7),(1,6),(1,8),(2,9),(2,14),(3,4),(3,8),(3,13),(4,10),(4,15),(5,6),(5,11),(6,12),(7,11),(7,14),(8,12),(9,10),(9,15),(10,13),(10,14),(11,12),(11,15),(12,13),(13,15),(14,15)],16)
=> ([(0,4),(0,5),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,11),(1,12),(1,13),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(2,14),(2,15),(3,4),(3,6),(3,7),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(4,5),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,6),(5,7),(5,8),(5,9),(5,10),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,14),(6,15),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,15),(9,10),(9,11),(9,13),(9,15),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14),(13,15),(14,15)],16)
=> ? = 1 + 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(0,15),(0,16),(1,7),(1,10),(2,8),(2,9),(3,4),(3,5),(3,6),(4,15),(4,16),(5,11),(5,15),(6,8),(6,11),(7,14),(7,16),(8,13),(9,10),(9,13),(10,14),(11,12),(11,13),(12,14),(12,15),(12,16),(13,14)],17)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,12),(1,13),(1,14),(1,15),(1,16),(2,4),(2,5),(2,7),(2,8),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(4,5),(4,6),(4,7),(4,8),(4,9),(4,11),(4,13),(4,15),(4,16),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,16),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,16),(10,11),(10,12),(10,14),(10,15),(10,16),(11,12),(11,14),(11,15),(11,16),(12,14),(12,15),(12,16),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 2 + 1
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ([(0,10),(0,11),(1,6),(1,7),(2,5),(2,9),(3,4),(3,8),(4,6),(4,13),(5,7),(5,14),(6,12),(7,12),(8,10),(8,13),(9,11),(9,14),(10,15),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,4),(1,5),(1,6),(1,7),(1,9),(1,11),(1,12),(1,13),(1,14),(1,15),(2,4),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,13),(4,14),(4,15),(5,6),(5,7),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,12),(6,14),(6,15),(7,8),(7,9),(7,11),(7,12),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,15),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 1 + 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(0,7),(0,15),(1,5),(1,14),(2,6),(2,8),(3,4),(3,13),(3,15),(4,6),(4,10),(5,12),(5,13),(6,11),(7,8),(7,9),(8,11),(9,11),(9,12),(9,15),(10,11),(10,12),(10,13),(12,14),(13,14),(14,15)],16)
=> ([(0,1),(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,11),(0,13),(0,14),(0,15),(1,4),(1,5),(1,6),(1,7),(1,10),(1,11),(1,12),(1,13),(1,14),(1,15),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(3,6),(3,7),(3,8),(3,9),(3,10),(3,12),(3,13),(3,14),(3,15),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,13),(4,14),(4,15),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(9,10),(9,12),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,15),(12,13),(12,14),(13,14),(13,15),(14,15)],16)
=> ? = 6 + 1
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ([(0,5),(0,14),(1,6),(1,7),(2,12),(2,13),(2,14),(3,6),(3,12),(3,13),(4,7),(4,11),(4,13),(5,8),(5,9),(6,10),(7,10),(8,11),(8,13),(8,14),(9,11),(9,12),(9,14),(10,11),(10,12)],15)
=> ([(0,1),(0,2),(0,4),(0,5),(0,6),(0,7),(0,8),(0,12),(0,13),(0,14),(1,3),(1,4),(1,6),(1,7),(1,8),(1,10),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(3,13),(3,14),(4,6),(4,7),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,12),(7,14),(8,9),(8,11),(8,12),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(13,14)],15)
=> ? = 3 + 1
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ([(0,6),(0,7),(1,8),(1,12),(2,5),(2,12),(3,5),(3,7),(3,14),(4,6),(4,11),(4,14),(5,13),(6,9),(7,9),(8,10),(8,13),(9,11),(9,14),(10,11),(10,12),(10,14),(11,13),(12,13),(13,14)],15)
=> ([(0,2),(0,5),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,3),(1,4),(1,6),(1,8),(1,10),(1,11),(1,12),(1,13),(1,14),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,4),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,12),(5,13),(5,14),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,10),(8,11),(8,12),(8,14),(9,10),(9,12),(9,13),(9,14),(10,11),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 1 + 1
001011 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> ([(0,8),(0,9),(1,7),(1,15),(2,6),(2,14),(3,14),(3,15),(4,6),(4,8),(4,12),(5,7),(5,9),(5,13),(6,16),(7,16),(8,10),(9,10),(10,12),(10,13),(11,12),(11,13),(11,14),(11,15),(12,16),(13,16),(14,16),(15,16)],17)
=> ([(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,3),(1,4),(1,7),(1,8),(1,9),(1,10),(1,12),(1,13),(1,14),(1,15),(1,16),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(2,11),(2,13),(2,14),(2,15),(2,16),(3,4),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,15),(5,6),(5,7),(5,8),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,16),(8,9),(8,10),(8,12),(8,13),(8,15),(8,16),(9,10),(9,11),(9,14),(9,15),(9,16),(10,12),(10,14),(10,15),(10,16),(11,12),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 1 + 1
001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,3),(2,4),(2,16),(3,6),(3,11),(4,5),(4,10),(5,8),(5,12),(6,9),(6,13),(7,12),(7,13),(7,15),(8,14),(9,14),(10,12),(10,15),(10,16),(11,13),(11,15),(11,16),(12,14),(13,14)],17)
=> ([(0,1),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,14),(0,15),(0,16),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,15),(1,16),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(3,4),(3,5),(3,6),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,16),(4,5),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(5,8),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,11),(8,12),(8,15),(8,16),(9,10),(9,11),(9,13),(9,15),(9,16),(10,12),(10,14),(10,15),(10,16),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 1 + 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St001878
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
00011 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 2
00111 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 3
01111 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,9),(2,16),(3,15),(4,12),(5,14),(6,7),(6,15),(7,5),(7,11),(8,1),(9,10),(10,3),(10,6),(11,14),(11,16),(12,8),(13,12),(14,13),(15,2),(15,11),(16,4),(16,13)],17)
=> ? = 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ?
=> ? = 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ?
=> ? = 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 4
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ?
=> ? = 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ?
=> ? = 2
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ?
=> ? = 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 6
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000098
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 1 + 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 1 + 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 2 + 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 1 + 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 2 + 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 1 + 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 1 + 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 1 + 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 1 + 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 1 + 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 1 + 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 1 + 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
00001 => ([(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)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 1 + 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ? = 1 + 1
00011 => ([(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)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 1 + 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,4),(2,11),(3,6),(3,9),(4,5),(4,8),(5,10),(6,10),(7,8),(7,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 3 + 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1 + 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 2 + 1
00111 => ([(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)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 1 + 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ? = 3 + 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ? = 2 + 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1 + 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1 + 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 1 + 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ? = 3 + 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,9),(0,11),(1,7),(1,8),(2,5),(2,9),(2,11),(3,4),(3,9),(3,11),(4,7),(4,10),(5,8),(5,10),(6,7),(6,8),(6,10),(10,11)],12)
=> ? = 3 + 1
01111 => ([(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)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 1 + 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 1 + 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,9),(0,11),(1,7),(1,8),(2,5),(2,9),(2,11),(3,4),(3,9),(3,11),(4,7),(4,10),(5,8),(5,10),(6,7),(6,8),(6,10),(10,11)],12)
=> ? = 1 + 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ? = 1 + 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 1 + 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 + 1
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1 + 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ? = 2 + 1
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ? = 1 + 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 1 + 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 1 + 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1 + 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,4),(2,11),(3,6),(3,9),(4,5),(4,8),(5,10),(6,10),(7,8),(7,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1 + 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,5),(0,9),(1,4),(1,9),(2,6),(2,8),(3,7),(3,8),(4,6),(4,10),(5,7),(5,10),(6,11),(7,11),(8,11),(9,10),(10,11)],12)
=> ? = 1 + 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ? = 1 + 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 1 + 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,3),(0,11),(1,2),(1,8),(2,9),(3,10),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(6,10),(7,11),(8,9),(10,11)],12)
=> ? = 1 + 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,9),(0,14),(1,7),(1,10),(2,8),(2,10),(3,4),(3,9),(3,14),(4,5),(4,11),(5,7),(5,13),(6,8),(6,13),(6,14),(7,12),(8,12),(9,11),(10,12),(11,13),(11,14),(12,13)],15)
=> ? = 1 + 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(0,7),(0,11),(1,6),(1,10),(2,8),(2,10),(3,9),(3,11),(4,8),(4,9),(4,14),(5,6),(5,7),(5,14),(6,12),(7,13),(8,12),(9,13),(10,12),(11,13),(12,14),(13,14)],15)
=> ? = 1 + 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(0,7),(0,15),(1,5),(1,14),(2,6),(2,8),(3,4),(3,13),(3,15),(4,6),(4,10),(5,12),(5,13),(6,11),(7,8),(7,9),(8,11),(9,11),(9,12),(9,15),(10,11),(10,12),(10,13),(12,14),(13,14),(14,15)],16)
=> ? = 4 + 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(0,5),(0,7),(1,6),(1,8),(2,9),(2,14),(3,4),(3,8),(3,13),(4,10),(4,15),(5,6),(5,11),(6,12),(7,11),(7,14),(8,12),(9,10),(9,15),(10,13),(10,14),(11,12),(11,15),(12,13),(13,15),(14,15)],16)
=> ? = 1 + 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(0,15),(0,16),(1,7),(1,10),(2,8),(2,9),(3,4),(3,5),(3,6),(4,15),(4,16),(5,11),(5,15),(6,8),(6,11),(7,14),(7,16),(8,13),(9,10),(9,13),(10,14),(11,12),(11,13),(12,14),(12,15),(12,16),(13,14)],17)
=> ? = 2 + 1
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ([(0,10),(0,11),(1,6),(1,7),(2,5),(2,9),(3,4),(3,8),(4,6),(4,13),(5,7),(5,14),(6,12),(7,12),(8,10),(8,13),(9,11),(9,14),(10,15),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(0,7),(0,15),(1,5),(1,14),(2,6),(2,8),(3,4),(3,13),(3,15),(4,6),(4,10),(5,12),(5,13),(6,11),(7,8),(7,9),(8,11),(9,11),(9,12),(9,15),(10,11),(10,12),(10,13),(12,14),(13,14),(14,15)],16)
=> ? = 6 + 1
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ([(0,5),(0,14),(1,6),(1,7),(2,12),(2,13),(2,14),(3,6),(3,12),(3,13),(4,7),(4,11),(4,13),(5,8),(5,9),(6,10),(7,10),(8,11),(8,13),(8,14),(9,11),(9,12),(9,14),(10,11),(10,12)],15)
=> ? = 3 + 1
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ([(0,6),(0,7),(1,8),(1,12),(2,5),(2,12),(3,5),(3,7),(3,14),(4,6),(4,11),(4,14),(5,13),(6,9),(7,9),(8,10),(8,13),(9,11),(9,14),(10,11),(10,12),(10,14),(11,13),(12,13),(13,14)],15)
=> ? = 1 + 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St001877
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 - 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 - 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1 - 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2 - 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 2 - 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 - 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 - 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 - 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 - 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1 - 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 - 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 - 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
00001 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1 - 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1 - 1
00011 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1 - 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3 - 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1 - 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 2 - 1
00111 => ([(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)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1 - 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 3 - 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2 - 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1 - 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1 - 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 3 - 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 3 - 1
01111 => ([(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)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1 - 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1 - 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? = 1 - 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1 - 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1 - 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 - 1
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 2 - 1
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1 - 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1 - 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? = 1 - 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1 - 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? = 1 - 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? = 1 - 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? = 1 - 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1 - 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,9),(2,16),(3,15),(4,12),(5,14),(6,7),(6,15),(7,5),(7,11),(8,1),(9,10),(10,3),(10,6),(11,14),(11,16),(12,8),(13,12),(14,13),(15,2),(15,11),(16,4),(16,13)],17)
=> ? = 1 - 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ?
=> ? = 1 - 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ?
=> ? = 1 - 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 4 - 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ?
=> ? = 1 - 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ?
=> ? = 2 - 1
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ?
=> ? = 1 - 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 6 - 1
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St000266
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 1
1 => ([(0,1)],2)
=> ([],2)
=> 1
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
00001 => ([(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)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 1
00011 => ([(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)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? = 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 2
00111 => ([(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)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 2
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
01111 => ([(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)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 2
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 2
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? = 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? = 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? = 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? = 4
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? = 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? = 2
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 1
Description
The number of spanning subgraphs of a graph with the same connected components.
A subgraph or factor of a graph is spanning, if it has the same vertex set [1]. The present statistic additionally requires the subgraph to have the same components. It can be obtained by evaluating the Tutte polynomial at the points $x=1$ and $y=2$, see [2,3].
By mistake, [2] refers to this statistic as the number of spanning subgraphs, which would be $2^m$, where $m$ is the number of edges. Equivalently, this would be the evaluation of the Tutte polynomial at $x=y=2$.
Matching statistic: St000267
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 1
1 => ([(0,1)],2)
=> ([],2)
=> 1
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
00001 => ([(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)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 1
00011 => ([(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)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? = 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 2
00111 => ([(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)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 2
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 3
01111 => ([(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)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 2
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? = 2
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? = 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? = 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? = 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? = 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? = 4
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? = 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? = 2
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 1
Description
The number of maximal spanning forests contained in a graph.
A maximal spanning forest in a graph is a maximal acyclic subgraph. In other words, a spanning forest is a union of spanning trees in all connected components. See also [1] for this and further definitions.
For connected graphs, this is the same as [[St000096]].
The following 127 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001029The size of the core of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001109The number of proper colourings of a graph with as few colours as possible. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001638The book thickness of a graph. St001654The monophonic hull number of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000260The radius of a connected graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000258The burning number of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000918The 2-limited packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001691The number of kings in a graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001765The number of connected components of the friends and strangers graph. St001261The Castelnuovo-Mumford regularity of a graph. St000264The girth of a graph, which is not a tree. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St000741The Colin de Verdière graph invariant. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001964The interval resolution global dimension of a poset. St001625The Möbius invariant of a lattice. St001060The distinguishing index of a graph.
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