Identifier
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Mp00136:
Binary words
—rotate back-to-front⟶
Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤ (values match St000439The position of the first down step of a Dyck path.)
Values
0 => 0 => [1] => [1,0] => 1
1 => 1 => [1] => [1,0] => 1
00 => 00 => [2] => [1,1,0,0] => 2
01 => 10 => [1,1] => [1,0,1,0] => 1
10 => 01 => [1,1] => [1,0,1,0] => 1
11 => 11 => [2] => [1,1,0,0] => 2
000 => 000 => [3] => [1,1,1,0,0,0] => 3
001 => 100 => [1,2] => [1,0,1,1,0,0] => 1
010 => 001 => [2,1] => [1,1,0,0,1,0] => 2
011 => 101 => [1,1,1] => [1,0,1,0,1,0] => 1
100 => 010 => [1,1,1] => [1,0,1,0,1,0] => 1
101 => 110 => [2,1] => [1,1,0,0,1,0] => 2
110 => 011 => [1,2] => [1,0,1,1,0,0] => 1
111 => 111 => [3] => [1,1,1,0,0,0] => 3
0000 => 0000 => [4] => [1,1,1,1,0,0,0,0] => 4
0001 => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 1
0010 => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 3
0011 => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 1
0100 => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
0101 => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 1
0110 => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2
0111 => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => 1
1000 => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => 1
1001 => 1100 => [2,2] => [1,1,0,0,1,1,0,0] => 2
1010 => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 1
1011 => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
1100 => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 1
1101 => 1110 => [3,1] => [1,1,1,0,0,0,1,0] => 3
1110 => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 1
1111 => 1111 => [4] => [1,1,1,1,0,0,0,0] => 4
00000 => 00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 5
00001 => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
00010 => 00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
00011 => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
00100 => 00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
00101 => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
00110 => 00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
00111 => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
01000 => 00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
01001 => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1
01010 => 00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 2
01011 => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1
01100 => 00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2
01101 => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1
01110 => 00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
01111 => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1
10000 => 01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1
10001 => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
10010 => 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1
10011 => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2
10100 => 01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1
10101 => 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 2
10110 => 01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1
10111 => 11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
11000 => 01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
11001 => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
11010 => 01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1
11011 => 11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3
11100 => 01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
11101 => 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
11110 => 01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
11111 => 11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => 5
000000 => 000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
000001 => 100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
000010 => 000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
000011 => 100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
000100 => 000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
000101 => 100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1
000110 => 000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
000111 => 100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1
001000 => 000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
001001 => 100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 1
001010 => 000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
001011 => 100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
001100 => 000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
001101 => 100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
001110 => 000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
001111 => 100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1
010000 => 001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
010001 => 101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
010010 => 001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
010011 => 101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1
010100 => 001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
010101 => 101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
010110 => 001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 2
010111 => 101011 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
011000 => 001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
011001 => 101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1
011010 => 001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
011011 => 101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
011100 => 001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
011101 => 101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 1
011110 => 001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
011111 => 101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1
100000 => 010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1
100001 => 110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
100010 => 010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 1
100011 => 110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
100100 => 010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
100101 => 110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
100110 => 010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1
>>> Load all 254 entries. <<<
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Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
In other words, this is the height of the first peak of $D$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
Map
rotate back-to-front
Description
The rotation of a binary word, last letter first.
This is the word obtained by moving the last letter to the beginnig.
This is the word obtained by moving the last letter to the beginnig.
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