Identifier
Values
0 => [2] => [1,1,0,0] => 1100 => 1
1 => [1,1] => [1,0,1,0] => 1010 => 2
00 => [3] => [1,1,1,0,0,0] => 111000 => 1
01 => [2,1] => [1,1,0,0,1,0] => 110010 => 2
10 => [1,2] => [1,0,1,1,0,0] => 101100 => 2
11 => [1,1,1] => [1,0,1,0,1,0] => 101010 => 3
000 => [4] => [1,1,1,1,0,0,0,0] => 11110000 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0] => 11100010 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 11001010 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0] => 10111000 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 10110010 => 3
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 10101100 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 10101010 => 4
=> [1] => [1,0] => 10 => 1
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Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's.
Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary word
Description
Return the Dyck word as binary word.
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.