Identifier
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Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤ
Values
0 => [2] => [1,1,0,0] => 1
1 => [1,1] => [1,0,1,0] => 2
00 => [3] => [1,1,1,0,0,0] => 1
01 => [2,1] => [1,1,0,0,1,0] => 2
10 => [1,2] => [1,0,1,1,0,0] => 2
11 => [1,1,1] => [1,0,1,0,1,0] => 3
000 => [4] => [1,1,1,1,0,0,0,0] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0] => 2
010 => [2,2] => [1,1,0,0,1,1,0,0] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 3
100 => [1,3] => [1,0,1,1,1,0,0,0] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 3
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 4
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 3
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 4
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 4
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 2
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 3
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 2
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 3
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 3
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 4
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 2
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 3
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 3
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 4
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 3
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 4
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 4
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 5
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 2
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 3
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 3
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 4
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 3
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 4
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 4
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 5
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 3
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 4
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 4
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 5
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 4
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 5
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 5
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 6
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 2
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 3
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 3
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 4
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 3
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 4
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 4
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Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
This is the number of points, excluding the origin, where the Dyck path has height 0.
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.
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.
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