Identifier
Values
0 => [2] => [1,1,0,0] => [2,1] => 2
1 => [1,1] => [1,0,1,0] => [1,2] => 1
00 => [3] => [1,1,1,0,0,0] => [3,2,1] => 2
01 => [2,1] => [1,1,0,0,1,0] => [2,1,3] => 3
10 => [1,2] => [1,0,1,1,0,0] => [1,3,2] => 2
11 => [1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 2
001 => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 3
100 => [1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 3
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 2
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 3
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 4
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 5
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 4
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 3
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 3
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 3
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 2
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 3
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 4
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 5
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 4
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 5
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 5
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 5
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 3
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 4
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 3
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 4
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 5
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 4
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 3
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 3
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 4
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 3
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 2
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 3
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 2
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 1
=> [1] => [1,0] => [1] => 1
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Description
The number of up-down runs of a permutation.
An up-down run of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$.
For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and
$643$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
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.