Identifier
Values
0 => [2] => [2] => [1,1,0,0] => 0
1 => [1,1] => [1,1] => [1,0,1,0] => 0
00 => [3] => [3] => [1,1,1,0,0,0] => 0
01 => [2,1] => [2,1] => [1,1,0,0,1,0] => 1
10 => [1,2] => [1,2] => [1,0,1,1,0,0] => 0
11 => [1,1,1] => [1,1,1] => [1,0,1,0,1,0] => 0
000 => [4] => [4] => [1,1,1,1,0,0,0,0] => 0
001 => [3,1] => [3,1] => [1,1,1,0,0,0,1,0] => 1
010 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 1
011 => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 1
100 => [1,3] => [1,3] => [1,0,1,1,1,0,0,0] => 0
101 => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 1
110 => [1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0] => 0
111 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 0
0000 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
0001 => [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
0010 => [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
0011 => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
0100 => [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
0101 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2
0110 => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 1
0111 => [2,1,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1
1000 => [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 0
1001 => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
1011 => [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
1100 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 0
1101 => [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
1110 => [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 0
1111 => [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 0
00000 => [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
00001 => [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
00010 => [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
00011 => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
00100 => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
00101 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 2
00110 => [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1
00111 => [3,1,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 1
01000 => [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
01001 => [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
01010 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
01011 => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
01100 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 1
01101 => [2,1,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
01110 => [2,1,1,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 1
01111 => [2,1,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
10000 => [1,5] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0
10001 => [1,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
10010 => [1,3,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 1
10011 => [1,3,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1
10100 => [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1
10101 => [1,2,2,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
10110 => [1,2,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 1
10111 => [1,2,1,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
11000 => [1,1,4] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0
11001 => [1,1,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
11010 => [1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1
11011 => [1,1,2,1,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
11100 => [1,1,1,3] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0
11101 => [1,1,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 1
11110 => [1,1,1,1,2] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0
11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
000000 => [7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
000001 => [6,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
000010 => [5,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 1
000011 => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 1
000100 => [4,3] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 1
000101 => [4,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 2
000110 => [4,1,2] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 1
000111 => [4,1,1,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 1
001000 => [3,4] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 1
001001 => [3,3,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 2
001010 => [3,2,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 2
001011 => [3,2,1,1] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 2
001100 => [3,1,3] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 1
001101 => [3,1,2,1] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 2
001110 => [3,1,1,2] => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 1
001111 => [3,1,1,1,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 1
010000 => [2,5] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 1
010001 => [2,4,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 2
010010 => [2,3,2] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 2
010011 => [2,3,1,1] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 2
010100 => [2,2,3] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 2
010101 => [2,2,2,1] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 3
010110 => [2,2,1,2] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 2
010111 => [2,2,1,1,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 2
011000 => [2,1,4] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 1
011001 => [2,1,3,1] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 2
011010 => [2,1,2,2] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 2
011011 => [2,1,2,1,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 2
011100 => [2,1,1,3] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 1
011101 => [2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 2
011110 => [2,1,1,1,2] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 1
011111 => [2,1,1,1,1,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 1
100000 => [1,6] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
100001 => [1,5,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 1
100010 => [1,4,2] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 1
100011 => [1,4,1,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 1
100100 => [1,3,3] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 1
100101 => [1,3,2,1] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 2
100110 => [1,3,1,2] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 1
>>> Load all 264 entries. <<<
100111 => [1,3,1,1,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 1
101000 => [1,2,4] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 1
101001 => [1,2,3,1] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 2
101010 => [1,2,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
101011 => [1,2,2,1,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 2
101100 => [1,2,1,3] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 1
101101 => [1,2,1,2,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 2
101110 => [1,2,1,1,2] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 1
101111 => [1,2,1,1,1,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 1
110000 => [1,1,5] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0
110001 => [1,1,4,1] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 1
110010 => [1,1,3,2] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 1
110011 => [1,1,3,1,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 1
110100 => [1,1,2,3] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 1
110101 => [1,1,2,2,1] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 2
110110 => [1,1,2,1,2] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 1
110111 => [1,1,2,1,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 1
111000 => [1,1,1,4] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0
111001 => [1,1,1,3,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 1
111010 => [1,1,1,2,2] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 1
111011 => [1,1,1,2,1,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 1
111100 => [1,1,1,1,3] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
111101 => [1,1,1,1,2,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 1
111110 => [1,1,1,1,1,2] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
0000000 => [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
0000001 => [7,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 1
0000010 => [6,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 1
0000011 => [6,1,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
0000110 => [5,1,2] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 1
0000111 => [5,1,1,1] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 1
0001000 => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => 1
0001010 => [4,2,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => 2
0001011 => [4,2,1,1] => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 2
0001100 => [4,1,3] => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 1
0001110 => [4,1,1,2] => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 1
0001111 => [4,1,1,1,1] => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 1
0010001 => [3,4,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => 2
0010011 => [3,3,1,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 2
0010101 => [3,2,2,1] => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0] => 3
0010110 => [3,2,1,2] => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0] => 2
0010111 => [3,2,1,1,1] => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 2
0011010 => [3,1,2,2] => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 2
0011011 => [3,1,2,1,1] => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 2
0011100 => [3,1,1,3] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 1
0011110 => [3,1,1,1,2] => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 1
0011111 => [3,1,1,1,1,1] => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 1
0100000 => [2,6] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
0100001 => [2,5,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => 2
0100010 => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => 2
0100011 => [2,4,1,1] => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 2
0100100 => [2,3,3] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => 2
0100110 => [2,3,1,2] => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0] => 2
0100111 => [2,3,1,1,1] => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 2
0101000 => [2,2,4] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => 2
0101001 => [2,2,3,1] => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => 3
0101010 => [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 3
0101011 => [2,2,2,1,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 3
0101100 => [2,2,1,3] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => 2
0101101 => [2,2,1,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => 3
0101110 => [2,2,1,1,2] => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 2
0101111 => [2,2,1,1,1,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 2
0110000 => [2,1,5] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => 1
0110001 => [2,1,4,1] => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => 2
0110011 => [2,1,3,1,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 2
0110100 => [2,1,2,3] => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0] => 2
0110101 => [2,1,2,2,1] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0] => 3
0110110 => [2,1,2,1,2] => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
0110111 => [2,1,2,1,1,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 2
0111000 => [2,1,1,4] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 1
0111001 => [2,1,1,3,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0] => 2
0111010 => [2,1,1,2,2] => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 2
0111011 => [2,1,1,2,1,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 2
0111100 => [2,1,1,1,3] => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 1
0111101 => [2,1,1,1,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => 2
0111110 => [2,1,1,1,1,2] => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 1
0111111 => [2,1,1,1,1,1,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 1
1000000 => [1,7] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
1000011 => [1,5,1,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 1
1000110 => [1,4,1,2] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 1
1000111 => [1,4,1,1,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 1
1001000 => [1,3,4] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 1
1001010 => [1,3,2,2] => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 2
1001011 => [1,3,2,1,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 2
1001110 => [1,3,1,1,2] => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 1
1001111 => [1,3,1,1,1,1] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 1
1010000 => [1,2,5] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 1
1010001 => [1,2,4,1] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0] => 2
1010011 => [1,2,3,1,1] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 2
1010100 => [1,2,2,3] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 2
1010101 => [1,2,2,2,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0] => 3
1010110 => [1,2,2,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
1010111 => [1,2,2,1,1,1] => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 2
1011000 => [1,2,1,4] => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0] => 1
1011001 => [1,2,1,3,1] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => 2
1011010 => [1,2,1,2,2] => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
1011011 => [1,2,1,2,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 2
1011100 => [1,2,1,1,3] => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 1
1011101 => [1,2,1,1,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => 2
1011110 => [1,2,1,1,1,2] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 1
1011111 => [1,2,1,1,1,1,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 1
1100000 => [1,1,6] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
1100011 => [1,1,4,1,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 1
1100100 => [1,1,3,3] => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 1
1100110 => [1,1,3,1,2] => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 1
1100111 => [1,1,3,1,1,1] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 1
1101000 => [1,1,2,4] => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 1
1101001 => [1,1,2,3,1] => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0] => 2
1101010 => [1,1,2,2,2] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
1101011 => [1,1,2,2,1,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 2
1101100 => [1,1,2,1,3] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => 1
1101101 => [1,1,2,1,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => 2
1101110 => [1,1,2,1,1,2] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 1
1101111 => [1,1,2,1,1,1,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 1
1110000 => [1,1,1,5] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0
1110010 => [1,1,1,3,2] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => 1
1110011 => [1,1,1,3,1,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 1
1110100 => [1,1,1,2,3] => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 1
1110101 => [1,1,1,2,2,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
1110110 => [1,1,1,2,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 1
1110111 => [1,1,1,2,1,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 1
1111000 => [1,1,1,1,4] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0
1111001 => [1,1,1,1,3,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 1
1111010 => [1,1,1,1,2,2] => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 1
1111011 => [1,1,1,1,2,1,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 1
1111100 => [1,1,1,1,1,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
1111101 => [1,1,1,1,1,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
1111110 => [1,1,1,1,1,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
00000000 => [9] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 0
00000001 => [8,1] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 1
00000010 => [7,2] => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => 1
00000011 => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 1
00000110 => [6,1,2] => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 1
00000111 => [6,1,1,1] => [1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
01000000 => [2,7] => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 1
01000001 => [2,6,1] => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2
10000000 => [1,8] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
10000011 => [1,6,1,1] => [1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => 1
10100000 => [1,2,6] => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
11000000 => [1,1,7] => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
11101110 => [1,1,1,2,1,1,2] => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 1
11110011 => [1,1,1,1,3,1,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 1
11110101 => [1,1,1,1,2,2,1] => [2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
11110110 => [1,1,1,1,2,1,2] => [2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 1
11111001 => [1,1,1,1,1,3,1] => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
11111010 => [1,1,1,1,1,2,2] => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 1
11111101 => [1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
11111110 => [1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
000000000 => [10] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 0
000000001 => [9,1] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => 1
000000010 => [8,2] => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => 1
000000011 => [8,1,1] => [1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 1
000000111 => [7,1,1,1] => [1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 1
010000000 => [2,8] => [2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 1
100000000 => [1,9] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 0
100000011 => [1,7,1,1] => [1,7,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => 1
110000000 => [1,1,8] => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
111110110 => [1,1,1,1,1,2,1,2] => [2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 1
111111001 => [1,1,1,1,1,1,3,1] => [3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
111111101 => [1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
111111110 => [1,1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
=> [1] => [1] => [1,0] => 0
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Description
The number of factors DDU in a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
Foata bijection
Description
The Foata bijection for compositions.
The Foata bijection $\phi$ is a bijection on the set of words whose letters are positive integers. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..