searching the database
Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000543
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00130: Permutations —descent tops⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 1
[2,1] => 1 => 1
[1,2,3] => 00 => 1
[1,3,2] => 01 => 2
[2,1,3] => 10 => 2
[2,3,1] => 01 => 2
[3,1,2] => 01 => 2
[3,2,1] => 11 => 1
[1,2,3,4] => 000 => 1
[1,2,4,3] => 001 => 3
[1,3,2,4] => 010 => 3
[1,3,4,2] => 001 => 3
[1,4,2,3] => 001 => 3
[1,4,3,2] => 011 => 3
[2,1,3,4] => 100 => 3
[2,1,4,3] => 101 => 3
[2,3,1,4] => 010 => 3
[2,3,4,1] => 001 => 3
[2,4,1,3] => 001 => 3
[2,4,3,1] => 011 => 3
[3,1,2,4] => 010 => 3
[3,1,4,2] => 011 => 3
[3,2,1,4] => 110 => 3
[3,2,4,1] => 011 => 3
[3,4,1,2] => 001 => 3
[3,4,2,1] => 101 => 3
[4,1,2,3] => 001 => 3
[4,1,3,2] => 011 => 3
[4,2,1,3] => 101 => 3
[4,2,3,1] => 011 => 3
[4,3,1,2] => 011 => 3
[4,3,2,1] => 111 => 1
[1,2,3,4,5] => 0000 => 1
[1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => 0010 => 4
[1,2,4,5,3] => 0001 => 4
[1,2,5,3,4] => 0001 => 4
[1,2,5,4,3] => 0011 => 4
[1,3,2,4,5] => 0100 => 4
[1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => 0010 => 4
[1,3,4,5,2] => 0001 => 4
[1,3,5,2,4] => 0001 => 4
[1,3,5,4,2] => 0011 => 4
[1,4,2,3,5] => 0010 => 4
[1,4,2,5,3] => 0011 => 4
[1,4,3,2,5] => 0110 => 4
[1,4,3,5,2] => 0011 => 4
[1,4,5,2,3] => 0001 => 4
[1,4,5,3,2] => 0101 => 2
Description
The size of the conjugacy class of a binary word.
Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
Matching statistic: St001633
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 30%
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 30%
Values
[1,2] => 0 => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => 1 => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => 00 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,3,4,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,3,2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[2,3,1,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[3,1,2,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,1,4] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,1,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,1,2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[4,3,2,1] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[1,2,4,3,5] => 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)
=> ? = 4 - 1
[1,2,4,5,3] => 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)
=> ? = 4 - 1
[1,2,5,3,4] => 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)
=> ? = 4 - 1
[1,2,5,4,3] => 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)
=> ? = 4 - 1
[1,3,2,4,5] => 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)
=> ? = 4 - 1
[1,3,2,5,4] => 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 - 1
[1,3,4,2,5] => 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)
=> ? = 4 - 1
[1,3,4,5,2] => 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)
=> ? = 4 - 1
[1,3,5,2,4] => 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)
=> ? = 4 - 1
[1,3,5,4,2] => 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)
=> ? = 4 - 1
[1,4,2,3,5] => 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)
=> ? = 4 - 1
[1,4,2,5,3] => 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)
=> ? = 4 - 1
[1,4,3,2,5] => 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)
=> ? = 4 - 1
[1,4,3,5,2] => 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)
=> ? = 4 - 1
[1,4,5,2,3] => 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)
=> ? = 4 - 1
[1,4,5,3,2] => 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 - 1
[1,5,2,3,4] => 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)
=> ? = 4 - 1
[1,5,2,4,3] => 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)
=> ? = 4 - 1
[1,5,3,2,4] => 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 - 1
[1,5,3,4,2] => 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)
=> ? = 4 - 1
[1,5,4,2,3] => 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)
=> ? = 4 - 1
[1,5,4,3,2] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,5,4] => 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)
=> ? = 4 - 1
[2,1,4,3,5] => 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 - 1
[2,1,4,5,3] => 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)
=> ? = 4 - 1
[2,1,5,3,4] => 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)
=> ? = 4 - 1
[2,1,5,4,3] => 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)
=> ? = 4 - 1
[2,3,1,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,1,5,4] => 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 - 1
[2,3,4,1,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,5,1] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,1,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,4,1] => 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)
=> ? = 4 - 1
[2,4,1,3,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,1,5,3] => 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)
=> ? = 4 - 1
[2,4,3,1,5] => 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)
=> ? = 4 - 1
[2,4,3,5,1] => 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)
=> ? = 4 - 1
[2,4,5,1,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,5,3,1] => 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 - 1
[2,5,1,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,5,1,4,3] => 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)
=> ? = 4 - 1
[2,5,3,1,4] => 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 - 1
[2,5,3,4,1] => 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)
=> ? = 4 - 1
[2,5,4,1,3] => 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)
=> ? = 4 - 1
[2,5,4,3,1] => 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)
=> ? = 4 - 1
[3,1,2,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[3,1,2,5,4] => 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 - 1
[3,1,4,2,5] => 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)
=> ? = 4 - 1
[5,4,3,2,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,4,5,6,7] => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[7,6,5,4,3,2,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000387
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => ([(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)
=> ? = 4 - 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,4,3] => ([(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)
=> ? = 4 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 - 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 2 - 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 2 - 1
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,5,4,3,2] => ([(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)
=> ? = 4 - 1
[2,1,3,4,5] => ([(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)
=> ? = 4 - 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 2 - 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,4,5,1] => ([(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)
=> ? = 4 - 1
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(2,3),(2,14),(3,13),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,9),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,14),(12,14),(13,14)],15)
=> ? = 2 - 1
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0 = 1 - 1
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0 = 1 - 1
Description
The matching number of a graph.
For a graph $G$, this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in $G$.
Matching statistic: St000848
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000848: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 30%
Mp00262: Binary words —poset of factors⟶ Posets
St000848: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 30%
Values
[1,2] => 0 => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => 1 => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => 00 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,3,4,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,3,2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[2,3,1,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[3,1,2,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,1,4] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,1,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,1,2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[4,3,2,1] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[1,2,4,3,5] => 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)
=> ? = 4 - 1
[1,2,4,5,3] => 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)
=> ? = 4 - 1
[1,2,5,3,4] => 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)
=> ? = 4 - 1
[1,2,5,4,3] => 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)
=> ? = 4 - 1
[1,3,2,4,5] => 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)
=> ? = 4 - 1
[1,3,2,5,4] => 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 - 1
[1,3,4,2,5] => 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)
=> ? = 4 - 1
[1,3,4,5,2] => 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)
=> ? = 4 - 1
[1,3,5,2,4] => 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)
=> ? = 4 - 1
[1,3,5,4,2] => 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)
=> ? = 4 - 1
[1,4,2,3,5] => 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)
=> ? = 4 - 1
[1,4,2,5,3] => 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)
=> ? = 4 - 1
[1,4,3,2,5] => 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)
=> ? = 4 - 1
[1,4,3,5,2] => 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)
=> ? = 4 - 1
[1,4,5,2,3] => 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)
=> ? = 4 - 1
[1,4,5,3,2] => 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 - 1
[1,5,2,3,4] => 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)
=> ? = 4 - 1
[1,5,2,4,3] => 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)
=> ? = 4 - 1
[1,5,3,2,4] => 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 - 1
[1,5,3,4,2] => 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)
=> ? = 4 - 1
[1,5,4,2,3] => 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)
=> ? = 4 - 1
[1,5,4,3,2] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,5,4] => 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)
=> ? = 4 - 1
[2,1,4,3,5] => 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 - 1
[2,1,4,5,3] => 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)
=> ? = 4 - 1
[2,1,5,3,4] => 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)
=> ? = 4 - 1
[2,1,5,4,3] => 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)
=> ? = 4 - 1
[2,3,1,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,1,5,4] => 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 - 1
[2,3,4,1,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,5,1] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,1,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,4,1] => 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)
=> ? = 4 - 1
[2,4,1,3,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,1,5,3] => 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)
=> ? = 4 - 1
[2,4,3,1,5] => 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)
=> ? = 4 - 1
[2,4,3,5,1] => 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)
=> ? = 4 - 1
[2,4,5,1,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,5,3,1] => 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 - 1
[2,5,1,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,5,1,4,3] => 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)
=> ? = 4 - 1
[2,5,3,1,4] => 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 - 1
[2,5,3,4,1] => 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)
=> ? = 4 - 1
[2,5,4,1,3] => 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)
=> ? = 4 - 1
[2,5,4,3,1] => 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)
=> ? = 4 - 1
[3,1,2,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[3,1,2,5,4] => 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 - 1
[3,1,4,2,5] => 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)
=> ? = 4 - 1
[5,4,3,2,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The balance constant multiplied with the number of linear extensions of a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$
Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced.
Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Matching statistic: St000849
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000849: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 30%
Mp00262: Binary words —poset of factors⟶ Posets
St000849: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 30%
Values
[1,2] => 0 => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => 1 => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => 00 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,3,4,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,3,2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[2,3,1,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[3,1,2,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,1,4] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,1,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,1,2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[4,3,2,1] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[1,2,4,3,5] => 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)
=> ? = 4 - 1
[1,2,4,5,3] => 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)
=> ? = 4 - 1
[1,2,5,3,4] => 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)
=> ? = 4 - 1
[1,2,5,4,3] => 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)
=> ? = 4 - 1
[1,3,2,4,5] => 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)
=> ? = 4 - 1
[1,3,2,5,4] => 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 - 1
[1,3,4,2,5] => 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)
=> ? = 4 - 1
[1,3,4,5,2] => 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)
=> ? = 4 - 1
[1,3,5,2,4] => 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)
=> ? = 4 - 1
[1,3,5,4,2] => 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)
=> ? = 4 - 1
[1,4,2,3,5] => 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)
=> ? = 4 - 1
[1,4,2,5,3] => 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)
=> ? = 4 - 1
[1,4,3,2,5] => 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)
=> ? = 4 - 1
[1,4,3,5,2] => 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)
=> ? = 4 - 1
[1,4,5,2,3] => 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)
=> ? = 4 - 1
[1,4,5,3,2] => 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 - 1
[1,5,2,3,4] => 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)
=> ? = 4 - 1
[1,5,2,4,3] => 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)
=> ? = 4 - 1
[1,5,3,2,4] => 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 - 1
[1,5,3,4,2] => 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)
=> ? = 4 - 1
[1,5,4,2,3] => 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)
=> ? = 4 - 1
[1,5,4,3,2] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,5,4] => 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)
=> ? = 4 - 1
[2,1,4,3,5] => 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 - 1
[2,1,4,5,3] => 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)
=> ? = 4 - 1
[2,1,5,3,4] => 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)
=> ? = 4 - 1
[2,1,5,4,3] => 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)
=> ? = 4 - 1
[2,3,1,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,1,5,4] => 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 - 1
[2,3,4,1,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,5,1] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,1,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,4,1] => 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)
=> ? = 4 - 1
[2,4,1,3,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,1,5,3] => 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)
=> ? = 4 - 1
[2,4,3,1,5] => 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)
=> ? = 4 - 1
[2,4,3,5,1] => 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)
=> ? = 4 - 1
[2,4,5,1,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,5,3,1] => 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 - 1
[2,5,1,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,5,1,4,3] => 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)
=> ? = 4 - 1
[2,5,3,1,4] => 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 - 1
[2,5,3,4,1] => 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)
=> ? = 4 - 1
[2,5,4,1,3] => 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)
=> ? = 4 - 1
[2,5,4,3,1] => 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)
=> ? = 4 - 1
[3,1,2,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[3,1,2,5,4] => 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 - 1
[3,1,4,2,5] => 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)
=> ? = 4 - 1
[5,4,3,2,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The number of 1/3-balanced pairs in a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$.
Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains.
Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St000985
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => ([(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)
=> ? = 4 - 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,4,3] => ([(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)
=> ? = 4 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 - 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 2 - 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 2 - 1
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,5,4,3,2] => ([(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)
=> ? = 4 - 1
[2,1,3,4,5] => ([(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)
=> ? = 4 - 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 2 - 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,4,5,1] => ([(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)
=> ? = 4 - 1
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(2,3),(2,14),(3,13),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,9),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,14),(12,14),(13,14)],15)
=> ? = 2 - 1
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0 = 1 - 1
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0 = 1 - 1
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001704
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[2,1] => ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 2 = 1 + 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 2 = 1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 2 = 1 + 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 2 = 1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 2 = 1 + 1
[1,2,3,5,4] => ([(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)
=> ? = 4 + 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,2,5,4,3] => ([(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)
=> ? = 4 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 + 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 + 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 2 + 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 2 + 1
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 + 1
[1,5,4,3,2] => ([(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)
=> ? = 4 + 1
[2,1,3,4,5] => ([(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)
=> ? = 4 + 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 + 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 + 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 + 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4 + 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 2 + 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[2,3,4,5,1] => ([(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)
=> ? = 4 + 1
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 + 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 2 + 1
[2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(2,3),(2,14),(3,13),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,9),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,14),(12,14),(13,14)],15)
=> ? = 2 + 1
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 2 = 1 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 2 = 1 + 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 2 = 1 + 1
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 2 = 1 + 1
Description
The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph.
The deck of a graph is the multiset of induced subgraphs obtained by deleting a single vertex.
The graph reconstruction conjecture states that the deck of a graph with at least three vertices determines the graph.
This statistic is only defined for graphs with at least two vertices, because there is only a single graph of the given size otherwise.
Matching statistic: St000362
Values
[1,2] => ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,4,3] => ([(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)
=> ([(2,7),(2,8),(3,5),(3,6),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(2,5),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,9),(2,11),(3,7),(3,8),(3,10),(3,11),(4,6),(4,8),(4,10),(4,11),(5,6),(5,7),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,11)],12)
=> ? = 4 - 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(2,9),(2,11),(2,12),(2,13),(3,8),(3,10),(3,12),(3,13),(4,7),(4,10),(4,12),(4,13),(5,6),(5,10),(5,12),(5,13),(6,9),(6,11),(6,12),(6,13),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,6),(2,8),(2,11),(3,5),(3,7),(3,10),(4,7),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 4 - 1
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,9),(2,11),(3,7),(3,8),(3,10),(3,11),(4,6),(4,8),(4,10),(4,11),(5,6),(5,7),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,11)],12)
=> ? = 4 - 1
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(2,9),(2,11),(2,12),(2,13),(3,8),(3,10),(3,12),(3,13),(4,7),(4,10),(4,12),(4,13),(5,6),(5,10),(5,12),(5,13),(6,9),(6,11),(6,12),(6,13),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,4),(3,5),(3,9),(3,10),(4,8),(4,9),(4,10),(5,7),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 - 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,6),(2,8),(2,11),(3,5),(3,7),(3,10),(4,7),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 4 - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 2 - 1
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[1,5,4,3,2] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,1,3,4,5] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ([(2,6),(2,7),(2,9),(3,4),(3,5),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4 - 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(2,5),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ([(2,7),(2,8),(3,5),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 - 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,4,5,1] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,4),(3,5),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,12),(6,9),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(2,9),(2,11),(2,12),(2,13),(3,8),(3,10),(3,12),(3,13),(4,7),(4,10),(4,12),(4,13),(5,6),(5,10),(5,12),(5,13),(6,9),(6,11),(6,12),(6,13),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ([(2,5),(2,6),(2,13),(3,8),(3,9),(3,12),(3,13),(4,10),(4,11),(4,12),(4,13),(5,7),(5,11),(5,12),(5,13),(6,7),(6,10),(6,12),(6,13),(7,8),(7,9),(7,12),(7,13),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,7),(3,8),(3,12),(4,5),(4,10),(4,11),(4,12),(5,9),(5,11),(5,12),(6,8),(6,10),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,6),(2,8),(2,11),(3,5),(3,7),(3,10),(4,7),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,4),(3,5),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,12),(6,9),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,7),(3,8),(3,12),(4,5),(4,10),(4,11),(4,12),(5,9),(5,11),(5,12),(6,8),(6,10),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(2,3),(2,14),(3,13),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,9),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,14),(12,14),(13,14)],15)
=> ([(2,10),(2,11),(2,14),(3,8),(3,9),(3,13),(4,9),(4,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,12),(5,13),(5,14),(6,8),(6,10),(6,12),(6,13),(6,14),(7,8),(7,11),(7,12),(7,13),(7,14),(8,13),(8,14),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14)],15)
=> ? = 2 - 1
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,4),(3,5),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,12),(6,9),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],6)
=> 0 = 1 - 1
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([],7)
=> 0 = 1 - 1
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([],7)
=> 0 = 1 - 1
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St001971
Values
[1,2] => ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,4,3] => ([(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)
=> ([(2,7),(2,8),(3,5),(3,6),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(2,5),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,9),(2,11),(3,7),(3,8),(3,10),(3,11),(4,6),(4,8),(4,10),(4,11),(5,6),(5,7),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,11)],12)
=> ? = 4 - 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(2,9),(2,11),(2,12),(2,13),(3,8),(3,10),(3,12),(3,13),(4,7),(4,10),(4,12),(4,13),(5,6),(5,10),(5,12),(5,13),(6,9),(6,11),(6,12),(6,13),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,6),(2,8),(2,11),(3,5),(3,7),(3,10),(4,7),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 4 - 1
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,9),(2,11),(3,7),(3,8),(3,10),(3,11),(4,6),(4,8),(4,10),(4,11),(5,6),(5,7),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,11)],12)
=> ? = 4 - 1
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(2,9),(2,11),(2,12),(2,13),(3,8),(3,10),(3,12),(3,13),(4,7),(4,10),(4,12),(4,13),(5,6),(5,10),(5,12),(5,13),(6,9),(6,11),(6,12),(6,13),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,4),(3,5),(3,9),(3,10),(4,8),(4,9),(4,10),(5,7),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 - 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,6),(2,8),(2,11),(3,5),(3,7),(3,10),(4,7),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 4 - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 2 - 1
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[1,5,4,3,2] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,1,3,4,5] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ([(2,6),(2,7),(2,9),(3,4),(3,5),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4 - 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(2,5),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,6),(4,7),(4,9),(5,6),(5,7),(5,9),(6,7),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ([(2,7),(2,8),(3,5),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(2,7),(2,8),(2,10),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 2 - 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,4,5,1] => ([(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)
=> ([(2,5),(2,7),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,4),(3,5),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,12),(6,9),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(2,8),(2,10),(3,6),(3,7),(3,9),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)
=> ? = 4 - 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(2,9),(2,11),(2,12),(2,13),(3,8),(3,10),(3,12),(3,13),(4,7),(4,10),(4,12),(4,13),(5,6),(5,10),(5,12),(5,13),(6,9),(6,11),(6,12),(6,13),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ([(2,5),(2,6),(2,13),(3,8),(3,9),(3,12),(3,13),(4,10),(4,11),(4,12),(4,13),(5,7),(5,11),(5,12),(5,13),(6,7),(6,10),(6,12),(6,13),(7,8),(7,9),(7,12),(7,13),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13)],14)
=> ? = 4 - 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,7),(3,8),(3,12),(4,5),(4,10),(4,11),(4,12),(5,9),(5,11),(5,12),(6,8),(6,10),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(2,6),(2,8),(2,11),(3,5),(3,7),(3,10),(4,7),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,4),(3,5),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,12),(6,9),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,7),(3,8),(3,12),(4,5),(4,10),(4,11),(4,12),(5,9),(5,11),(5,12),(6,8),(6,10),(6,11),(6,12),(7,9),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(2,3),(2,14),(3,13),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,9),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,14),(12,14),(13,14)],15)
=> ([(2,10),(2,11),(2,14),(3,8),(3,9),(3,13),(4,9),(4,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,12),(5,13),(5,14),(6,8),(6,10),(6,12),(6,13),(6,14),(7,8),(7,11),(7,12),(7,13),(7,14),(8,13),(8,14),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14)],15)
=> ? = 2 - 1
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(2,8),(2,9),(2,11),(3,6),(3,10),(3,12),(4,9),(4,10),(4,11),(4,12),(5,8),(5,10),(5,11),(5,12),(6,7),(6,11),(6,12),(7,8),(7,9),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(2,10),(2,11),(2,12),(3,4),(3,5),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,12),(6,9),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12)],13)
=> ? = 4 - 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(2,7),(2,9),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],6)
=> 0 = 1 - 1
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([],7)
=> 0 = 1 - 1
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([],7)
=> 0 = 1 - 1
Description
The number of negative eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001812
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 2 = 3 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => ([(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)
=> ? = 4 - 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,2,5,4,3] => ([(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)
=> ? = 4 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 - 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 2 - 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 2 - 1
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[1,5,4,3,2] => ([(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)
=> ? = 4 - 1
[2,1,3,4,5] => ([(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)
=> ? = 4 - 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4 - 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 2 - 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[2,3,4,5,1] => ([(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)
=> ? = 4 - 1
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 - 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 - 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ? = 4 - 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!