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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000295
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St000295: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 1
01 => 0
10 => 0
11 => 1
000 => 2
001 => 0
010 => 1
011 => 0
100 => 0
101 => 1
110 => 0
111 => 2
0000 => 3
0001 => 0
0010 => 1
0011 => 0
0100 => 1
0101 => 2
0110 => 1
0111 => 0
1000 => 0
1001 => 1
1010 => 2
1011 => 1
1100 => 0
1101 => 1
1110 => 0
1111 => 3
00000 => 4
00001 => 0
00010 => 1
00011 => 0
00100 => 2
00101 => 0
00110 => 1
00111 => 0
01000 => 1
01001 => 2
01010 => 3
01011 => 0
01100 => 1
01101 => 2
01110 => 1
01111 => 0
10000 => 0
10001 => 1
10010 => 2
10011 => 1
Description
The length of the border of a binary word.
The border of a word is the longest word which is both a proper prefix and a proper suffix, including a possible empty border.
Matching statistic: St001877
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(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: 44%
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: 44%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
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: St001052
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001052: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001052: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,2] => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 2 = 1 + 1
01 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 2 = 1 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3 = 2 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1 = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 2 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 4 = 3 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 3 = 2 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 2 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1 = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 3 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 5 = 4 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,5,1,2,3] => 1 = 0 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => 3 = 2 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => 4 = 3 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => 1 = 0 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => 3 = 2 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1 = 0 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => 2 = 1 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => 3 = 2 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => 2 = 1 + 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,5,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,3,4,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [7,8,6,1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5} + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,1,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5} + 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,1,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5} + 1
Description
The length of the exterior of a permutation.
The '''exterior''' of a permutation is the longest proper prefix that is also a suffix, when viewed as a pattern. In other words, the length of the exterior of a permutation $\sigma$ of length $n$ is the largest $i < n$ such that the first $i$ entries of $\sigma$ are in the same relative order as the last $i$ entries of $\sigma$.
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: 33%
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: 33%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,1,1}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,1,1,2,2}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
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)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4}
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: St000771
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Values
0 => [1] => ([],1)
=> 1 = 0 + 1
1 => [1] => ([],1)
=> 1 = 0 + 1
00 => [2] => ([],2)
=> ? ∊ {1,1} + 1
01 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
10 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
11 => [2] => ([],2)
=> ? ∊ {1,1} + 1
000 => [3] => ([],3)
=> ? ∊ {0,0,2,2} + 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,2,2} + 1
100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,2,2} + 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
111 => [3] => ([],3)
=> ? ∊ {0,0,2,2} + 1
0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
1111 => [4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,3,3} + 1
00000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
10000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
11000 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
11111 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,4,4} + 1
000000 => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
000101 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
001001 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
001010 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
001101 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
010010 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
010100 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
010110 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
011010 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
011101 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
100000 => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
100010 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,5,5} + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Values
0 => [1] => ([],1)
=> 1 = 0 + 1
1 => [1] => ([],1)
=> 1 = 0 + 1
00 => [2] => ([],2)
=> ? ∊ {1,1} + 1
01 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
10 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
11 => [2] => ([],2)
=> ? ∊ {1,1} + 1
000 => [3] => ([],3)
=> ? ∊ {0,0,2,2} + 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,2,2} + 1
100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,2,2} + 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
111 => [3] => ([],3)
=> ? ∊ {0,0,2,2} + 1
0000 => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
1111 => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3,3} + 1
00000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
10000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
11000 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
11111 => [5] => ([],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,4,4} + 1
000000 => [6] => ([],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
000101 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
001001 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
001010 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
001101 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
010010 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
010100 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
010110 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
011010 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
011101 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
100000 => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
100010 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,5,5} + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 78%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 78%
Values
0 => [1] => [1,0]
=> [1,0]
=> 0
1 => [1] => [1,0]
=> [1,0]
=> 0
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,0}
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,0}
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2}
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
101011 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
0010101 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
0100101 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
0101001 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
0101010 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
0101011 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
0101101 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4
0110101 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
1001010 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 78%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 78%
Values
0 => 1 => [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
1 => 1 => [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
00 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
01 => 10 => [1,2] => ([(1,2)],3)
=> 2 = 0 + 2
10 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
000 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
001 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
010 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1} + 2
011 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1} + 2
100 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1} + 2
101 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
110 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
0000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
0001 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
0010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
0011 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
0100 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
0110 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
0111 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
1000 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
1001 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
1010 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
1011 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
1100 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2} + 2
1101 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
1110 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
1111 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
00000 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
00001 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
00010 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
00011 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
00100 => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
00101 => 01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
00110 => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
00111 => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01000 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01010 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01011 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01100 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01101 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01110 => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
01111 => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10000 => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10001 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10010 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10011 => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10100 => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10101 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10110 => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
10111 => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
11000 => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
11001 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
11010 => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
11011 => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
11100 => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3} + 2
11101 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
11110 => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 4 + 2
11111 => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 4 + 2
000000 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
000001 => 000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
000010 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
000011 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
000100 => 001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
000101 => 001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
000110 => 001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
000111 => 001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
001000 => 010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
001001 => 010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 2
111101 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 4 + 2
111110 => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 5 + 2
111111 => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 5 + 2
0000000 => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
1111110 => 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8 = 6 + 2
1111111 => 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8 = 6 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000469
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 2 = 0 + 2
1 => ([(0,1)],2)
=> ([],2)
=> 2 = 0 + 2
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 1 + 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2 = 0 + 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2 = 0 + 2
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 1 + 2
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 2 + 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 1 + 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 1 + 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 2 + 2
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 3 + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
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)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 3 + 2
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 4 + 2
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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 2
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 4 + 2
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7 = 5 + 2
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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 2
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7 = 5 + 2
Description
The distinguishing number of a graph.
This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring.
For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].
Matching statistic: St000776
Values
0 => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
1 => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 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)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2} + 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ([(0,1),(0,2),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(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)
=> ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ([(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)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(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)
=> ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(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)
=> ([(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ([(0,1),(0,4),(0,6),(0,7),(0,9),(0,10),(0,11),(1,3),(1,6),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 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)
=> ([(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)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ([(0,2),(0,8),(0,9),(0,10),(0,11),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ([(0,1),(0,2),(0,6),(0,7),(0,9),(0,10),(0,11),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ([(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,6),(1,8),(1,9),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,4),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ([(0,3),(0,5),(0,7),(0,9),(0,10),(0,11),(1,2),(1,4),(1,6),(1,8),(1,10),(1,11),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ([(0,3),(0,9),(0,10),(0,12),(0,13),(0,14),(1,2),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,6),(2,7),(2,8),(2,11),(2,12),(2,13),(2,14),(3,5),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,11),(8,12),(8,13),(8,14),(9,10),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ([(0,3),(0,8),(0,10),(0,12),(0,13),(0,14),(1,2),(1,7),(1,9),(1,11),(1,13),(1,14),(2,5),(2,7),(2,9),(2,11),(2,12),(2,13),(2,14),(3,6),(3,8),(3,10),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ([(0,2),(0,9),(0,12),(0,13),(0,14),(0,15),(1,5),(1,6),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(2,4),(2,8),(2,9),(2,11),(2,12),(2,13),(2,14),(2,15),(3,5),(3,6),(3,7),(3,8),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,6),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,12),(9,13),(9,14),(9,15),(10,11),(10,13),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(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)
=> ([(0,3),(0,4),(0,12),(0,13),(0,14),(0,15),(1,2),(1,7),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(2,5),(2,7),(2,9),(2,10),(2,11),(2,13),(2,14),(2,15),(3,4),(3,6),(3,9),(3,11),(3,12),(3,13),(3,14),(3,15),(4,6),(4,8),(4,10),(4,12),(4,13),(4,14),(4,15),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,11),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ([(0,4),(0,5),(0,11),(0,14),(0,15),(0,16),(1,2),(1,8),(1,12),(1,13),(1,14),(1,15),(1,16),(2,3),(2,8),(2,12),(2,13),(2,14),(2,15),(2,16),(3,7),(3,8),(3,10),(3,11),(3,12),(3,13),(3,15),(3,16),(4,5),(4,6),(4,9),(4,11),(4,13),(4,14),(4,15),(4,16),(5,6),(5,10),(5,11),(5,12),(5,14),(5,15),(5,16),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,9),(8,12),(8,13),(8,14),(8,15),(8,16),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4} + 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
Description
The maximal multiplicity of an eigenvalue in a graph.
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