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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St001414
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 0
11 => 0
000 => 1
001 => 0
010 => 1
011 => 0
100 => 0
101 => 1
110 => 0
111 => 1
0000 => 1
0001 => 1
0010 => 0
0011 => 0
0100 => 1
0101 => 1
0110 => 0
0111 => 0
1000 => 0
1001 => 0
1010 => 1
1011 => 1
1100 => 0
1101 => 0
1110 => 1
1111 => 1
00000 => 2
00001 => 1
00010 => 1
00011 => 1
00100 => 2
00101 => 0
00110 => 0
00111 => 0
01000 => 1
01001 => 1
01010 => 2
01011 => 1
01100 => 0
01101 => 0
01110 => 2
01111 => 0
10000 => 0
10001 => 2
10010 => 0
10011 => 0
Description
Half the length of the longest odd length palindromic prefix of a binary word.
More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
Matching statistic: St001877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 80%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 80%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
000111 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011000 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
011100 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100011 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
100111 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
110001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
110011 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
111000 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
111001 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0000110 => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 0
0000111 => [4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> 0
0001000 => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0001100 => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001110 => [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
0001111 => [3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
0010001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
0010010 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
0010011 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
0010110 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011000 => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011001 => [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 2
0011010 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
0011011 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
0011100 => [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011110 => [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100010 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0100011 => [1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100100 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
000111 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011000 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
011100 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100011 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
100111 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
110001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
110011 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
111000 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
111001 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0000110 => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 0
0000111 => [4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> 0
0001000 => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0001100 => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001110 => [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
0001111 => [3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
0010010 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
0010110 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011000 => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011010 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
0011100 => [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011110 => [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100010 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0100011 => [1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100100 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
0100101 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 0
0100110 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
0100111 => [1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
0101100 => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001556
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
0 => [2] => [1,1,0,0]
=> [1,2] => 0
1 => [1,1] => [1,0,1,0]
=> [2,1] => 0
00 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,5,3,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,4,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001630
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0} + 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0} + 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2 = 1 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2 = 1 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 2 = 1 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2 = 1 + 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1} + 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,10),(6,3),(6,11),(7,2),(7,11),(8,9),(9,4),(9,5),(10,6),(10,7),(11,12),(11,13),(12,14),(13,14),(14,1)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,9),(2,14),(3,13),(4,13),(4,14),(5,16),(6,16),(7,11),(8,12),(9,5),(9,6),(10,11),(10,12),(11,15),(12,15),(13,7),(13,10),(14,8),(14,10),(15,1),(16,2),(16,3),(16,4)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,8),(1,11),(2,11),(3,11),(4,10),(5,10),(7,9),(8,7),(9,1),(9,2),(9,3),(10,6),(11,4),(11,5)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,9),(1,11),(2,10),(3,14),(5,13),(6,12),(7,1),(7,13),(8,2),(8,14),(9,3),(9,8),(10,7),(11,12),(12,4),(13,6),(13,11),(14,5),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,8),(2,13),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,12),(8,9),(9,5),(9,6),(10,13),(11,7),(12,2),(12,10),(13,1)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,9),(2,16),(3,15),(4,12),(5,14),(6,7),(6,15),(7,5),(7,11),(8,1),(9,10),(10,3),(10,6),(11,14),(11,16),(12,8),(13,12),(14,13),(15,2),(15,11),(16,4),(16,13)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001306
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 0
1 => ([(0,1)],2)
=> ([],2)
=> 0
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
The number of induced paths on four vertices in a graph.
Matching statistic: St001350
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 0
1 => ([(0,1)],2)
=> ([],2)
=> 0
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
Half of the Albertson index of a graph.
This is $\frac{1}{2}\sum_{\{u,v\}\in E}|d(u)-d(v)|$, where $E$ is the set of edges and $d_v$ is the degree of vertex $v$, see [1].
In particular, this statistic vanishes on graphs whose components are all regular, see [2].
Matching statistic: St001353
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 0
1 => ([(0,1)],2)
=> ([],2)
=> 0
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
The number of prime nodes in the modular decomposition of a graph.
Matching statistic: St001723
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 0
1 => ([(0,1)],2)
=> ([],2)
=> 0
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
The differential of a graph.
The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$.
The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$.
The differential of a graph is the maximal differential of a set of vertices.
Matching statistic: St001724
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 0
1 => ([(0,1)],2)
=> ([],2)
=> 0
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
The 2-packing differential of a graph.
The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$.
The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$.
A set $S\subseteq V(G)$ is $2$-packing if the closed neighbourhoods of any two vertices in $S$ have empty intersection.
The $2$-packing differential of a graph is the maximal differential of any $2$-packing set of vertices.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000535The rank-width of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001393The induced matching number of a graph. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000985The number of positive eigenvalues of the adjacency matrix of the graph.
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