Identifier
-
Mp00104:
Binary words
—reverse⟶
Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤ
Values
0 => 0 => [2] => [1,1,0,0] => 0
1 => 1 => [1,1] => [1,0,1,0] => 1
00 => 00 => [3] => [1,1,1,0,0,0] => 0
01 => 10 => [1,2] => [1,0,1,1,0,0] => 1
10 => 01 => [2,1] => [1,1,0,0,1,0] => 2
11 => 11 => [1,1,1] => [1,0,1,0,1,0] => 2
000 => 000 => [4] => [1,1,1,1,0,0,0,0] => 0
001 => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1
010 => 010 => [2,2] => [1,1,0,0,1,1,0,0] => 2
011 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2
100 => 001 => [3,1] => [1,1,1,0,0,0,1,0] => 3
101 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
110 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 3
111 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3
0000 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
0001 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
0010 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 2
0100 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3
0110 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
0111 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3
1000 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
1001 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4
1011 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4
1101 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
1110 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4
1111 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4
00000 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
00001 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
00010 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
00011 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
00100 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
00101 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
00110 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
00111 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
01000 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
01001 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4
01010 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
01011 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
01100 => 00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4
01101 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
01110 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
01111 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
10000 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
10001 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
10010 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 5
10011 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5
10100 => 00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 5
10101 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
10110 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 5
10111 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
11000 => 00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 5
11001 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 5
11010 => 01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 5
11011 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
11100 => 00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 5
11101 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5
11110 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 5
11111 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
000000 => 000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
000001 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
000010 => 010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 2
000011 => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 2
000100 => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 3
000101 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 3
000110 => 011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 3
000111 => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 3
001000 => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 4
001001 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 4
001010 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 4
001011 => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 4
001100 => 001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 4
001101 => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 4
001110 => 011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 4
001111 => 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 4
010000 => 000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 5
010001 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 5
010010 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5
010011 => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 5
010100 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 5
010101 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 5
010110 => 011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 5
010111 => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 5
011000 => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 5
011001 => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 5
011010 => 010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 5
011011 => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 5
011100 => 001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 5
011101 => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 5
011110 => 011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 5
011111 => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 5
100000 => 000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6
100001 => 100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
100010 => 010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 6
100011 => 110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 6
100100 => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 6
100101 => 101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
100110 => 011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 6
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Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$ \sum_v (j_v-i_v)/2. $$
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$ \sum_v (j_v-i_v)/2. $$
Map
reverse
Description
Return the reversal of a binary word.
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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