Identifier
Values
0 => [2] => [1,1,0,0] => [] => 0
1 => [1,1] => [1,0,1,0] => [1] => 0
00 => [3] => [1,1,1,0,0,0] => [] => 0
01 => [2,1] => [1,1,0,0,1,0] => [2] => 0
10 => [1,2] => [1,0,1,1,0,0] => [1,1] => 1
11 => [1,1,1] => [1,0,1,0,1,0] => [2,1] => 0
000 => [4] => [1,1,1,1,0,0,0,0] => [] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [3] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [5,4,3] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 1
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 2
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 2
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 2
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 1
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 1
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 2
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 1
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 1
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 1
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 0
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [] => 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => 0
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,5] => 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [6,5] => 0
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [4,4,4] => 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [6,4,4] => 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [5,5,4] => 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [6,5,4] => 0
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3] => 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [6,3,3,3] => 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [5,5,3,3] => 2
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [6,5,3,3] => 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [4,4,4,3] => 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [6,4,4,3] => 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [5,5,4,3] => 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3] => 0
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [6,2,2,2,2] => 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [5,5,2,2,2] => 2
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,2,2] => 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [4,4,4,2,2] => 2
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [6,4,4,2,2] => 2
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [5,5,4,2,2] => 2
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,2] => 1
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2] => 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [6,3,3,3,2] => 1
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [5,5,3,3,2] => 2
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,3,2] => 1
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [4,4,4,3,2] => 1
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [6,4,4,3,2] => 1
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [5,5,4,3,2] => 1
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2] => 0
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,1,1,1,1] => 1
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [5,5,1,1,1,1] => 2
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,1,1,1] => 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [4,4,4,1,1,1] => 2
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,4,1,1,1] => 2
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [5,5,4,1,1,1] => 2
>>> Load all 177 entries. <<<
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,1,1] => 1
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [3,3,3,3,1,1] => 2
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,3,3,1,1] => 2
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [5,5,3,3,1,1] => 3
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,3,1,1] => 2
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [4,4,4,3,1,1] => 2
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,4,3,1,1] => 2
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [5,5,4,3,1,1] => 2
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,1] => 1
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => 1
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,2,2,2,1] => 1
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [5,5,2,2,2,1] => 2
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,2,2,1] => 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [4,4,4,2,2,1] => 2
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,4,2,2,1] => 2
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [5,5,4,2,2,1] => 2
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,2,1] => 1
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [3,3,3,3,2,1] => 1
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,3,3,2,1] => 1
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [5,5,3,3,2,1] => 2
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,3,2,1] => 1
111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [4,4,4,3,2,1] => 1
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,4,3,2,1] => 1
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [5,5,4,3,2,1] => 1
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 0
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [] => 0
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7] => 0
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => [6,6] => 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => [7,6] => 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => [5,5,5] => 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [4,4,4,4] => 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0] => [6,6,5,4] => 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => [3,3,3,3,3] => 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0] => [4,4,4,4,3] => 1
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2] => 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [6,6,2,2,2,2] => 2
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => [7,6,2,2,2,2] => 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => [3,3,3,3,3,2] => 1
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,2] => 0
1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 1
1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,2,2,1] => 1
1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [4,4,4,4,3,2,1] => 1
1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [5,5,5,4,3,2,1] => 1
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [6,6,5,4,3,2,1] => 1
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [7,6,5,4,3,2,1] => 0
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [] => 0
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8] => 0
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => [7,7] => 1
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [8,7] => 0
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0] => [6,6,6] => 1
00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0] => [8,6,6] => 1
00001000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0] => [5,5,5,5] => 1
00010000 => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [4,4,4,4,4] => 1
00100000 => [3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3,3] => 1
01000000 => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,2,2,2,2,2,2] => 1
01000001 => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [8,2,2,2,2,2,2] => 1
01100000 => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [3,3,3,3,3,3,2] => 1
10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1] => 1
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [6,6,6,5,4,3,2,1] => 1
11111110 => [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] => [7,7,6,5,4,3,2,1] => 1
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,3,2,1] => 0
000000000 => [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] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [9] => 0
000000010 => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => [8,8] => 1
000000011 => [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => [9,8] => 0
000001000 => [6,4] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0] => [6,6,6,6] => 1
000010000 => [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [5,5,5,5,5] => 1
000100000 => [4,6] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [4,4,4,4,4,4] => 1
001000000 => [3,7] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [3,3,3,3,3,3,3] => 1
010000000 => [2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2,2,2,2,2,2] => 1
100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1] => 1
111111110 => [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] => [8,8,7,6,5,4,3,2,1] => 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,8,7,6,5,4,3,2,1] => 0
1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,1,1,1] => 1
=> [1] => [1,0] => [] => 0
0000000001 => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [10] => 0
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Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
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
bounce path
Description
The bounce path determined by an integer composition.