Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000682
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Mapping
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 1
10 => 0
11 => 0
000 => 0
001 => 2
010 => 1
011 => 2
100 => 0
101 => 1
110 => 0
111 => 0
0000 => 0
0001 => 3
0010 => 2
0011 => 0
0100 => 1
0101 => 1
0110 => 2
0111 => 3
1000 => 0
1001 => 2
1010 => 1
1011 => 2
1100 => 0
1101 => 1
1110 => 0
1111 => 0
00000 => 0
00001 => 4
00010 => 3
00011 => 6
00100 => 2
00101 => 5
00110 => 0
00111 => 6
01000 => 1
01001 => 4
01010 => 1
01011 => 5
01100 => 2
01101 => 4
01110 => 3
01111 => 4
10000 => 0
10001 => 3
10010 => 2
10011 => 0
Description
The Grundy value of Welter's game on a binary word. Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
Matching statistic: St000772
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 31%
Values
0 => 0 => [1] => ([],1)
 => 1 = 0 + 1
1 => 1 => [1] => ([],1)
 => 1 = 0 + 1
00 => 00 => [2] => ([],2)
 => ? ∊ {0,1} + 1
01 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
10 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
11 => 11 => [2] => ([],2)
 => ? ∊ {0,1} + 1
000 => 000 => [3] => ([],3)
 => ? ∊ {0,1,1,2,2} + 1
001 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
010 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
011 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,2,2} + 1
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
101 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,2,2} + 1
110 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,1,1,2,2} + 1
111 => 111 => [3] => ([],3)
 => ? ∊ {0,1,1,2,2} + 1
0000 => 0000 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
0001 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
0010 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
0011 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
0100 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
0101 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
0110 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
0111 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
1001 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
1010 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
1011 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
1100 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
1101 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
1110 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
1111 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,2,2,2,3,3} + 1
00000 => 00000 => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
00001 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
00010 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
00011 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
00100 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
00101 => 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 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
00111 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
01000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
01001 => 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
01010 => 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
01011 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
01100 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
01101 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
01110 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
01111 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
10000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
10001 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
10010 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
10011 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
10100 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
10101 => 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
10110 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
10111 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11000 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11001 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11010 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11011 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11100 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11101 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11110 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
11111 => 11111 => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,4,4,4,5,5,6,6} + 1
000000 => 000000 => [6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
000001 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
000010 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
000011 => 000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
000100 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
000101 => 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 => 000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
000111 => 000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
001000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
001001 => 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 => 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
001011 => 001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
001100 => 000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
001101 => 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 => 000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
001111 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7} + 1
010000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
010001 => 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
010010 => 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
010011 => 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
010100 => 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
010101 => 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
100000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
0000001 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
0000010 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
0000100 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
0000101 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
0001000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
0001001 => 0001001 => [3,1,2,1] => ([(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)
 => 1 = 0 + 1
0001010 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
0001101 => 0001101 => [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)
 => 1 = 0 + 1
0010000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
0010001 => 0001001 => [3,1,2,1] => ([(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)
 => 1 = 0 + 1
0010010 => 0001001 => [3,1,2,1] => ([(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)
 => 1 = 0 + 1
0010100 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
0010101 => 0010101 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
0011010 => 0001101 => [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)
 => 1 = 0 + 1
0011101 => 0011101 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 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].