Identifier
-
Mp00105:
Binary words
—complement⟶
Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤ
Values
0 => 1 => [1,1] => [1,0,1,0] => 1
1 => 0 => [2] => [1,1,0,0] => 2
00 => 11 => [1,1,1] => [1,0,1,0,1,0] => 1
01 => 10 => [1,2] => [1,0,1,1,0,0] => 2
10 => 01 => [2,1] => [1,1,0,0,1,0] => 2
11 => 00 => [3] => [1,1,1,0,0,0] => 3
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 2
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 3
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0] => 2
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0] => 3
111 => 000 => [4] => [1,1,1,1,0,0,0,0] => 4
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 3
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 2
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 5
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 2
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 3
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 2
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
10100 => 01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
10101 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
10110 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 3
10111 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
11000 => 00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
11001 => 00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
11010 => 00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
11011 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
11100 => 00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
11101 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
11110 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
11111 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
000000 => 111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
000001 => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
000010 => 111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
000011 => 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
000100 => 111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 2
000101 => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
000110 => 111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 3
000111 => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
001000 => 110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 2
001001 => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
001010 => 110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 2
001011 => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 3
001100 => 110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 3
001101 => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 3
001110 => 110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 4
001111 => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 5
010000 => 101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 2
010001 => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
010010 => 101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 2
010011 => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 3
010100 => 101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 2
010101 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
010110 => 101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 3
010111 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 4
011000 => 100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 3
011001 => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 3
011010 => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 3
011011 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 3
011100 => 100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 4
011101 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 4
011110 => 100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 5
011111 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
100000 => 011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 2
100001 => 011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 2
100010 => 011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 2
100011 => 011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 3
100100 => 011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 2
100101 => 011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 2
100110 => 011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 3
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Description
The length of the maximal rise of a Dyck path.
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.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
complement
Description
Send a binary word to the word obtained by interchanging the two letters.
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