Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000834: Permutations ⟶ ℤ
Values
0 => [2] => [1,1,0,0] => [2,1] => 0
1 => [1,1] => [1,0,1,0] => [1,2] => 1
00 => [3] => [1,1,1,0,0,0] => [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0] => [2,1,3] => 1
10 => [1,2] => [1,0,1,1,0,0] => [1,3,2] => 1
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] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
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] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
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] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 2
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 2
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 3
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 2
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 2
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 2
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 2
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
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
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [7,6,5,4,3,2,1] => 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6,5,4,3,2,1,7] => 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [5,4,3,2,1,7,6] => 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [5,4,3,2,1,6,7] => 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [4,3,2,1,5,6,7] => 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [3,2,1,4,5,6,7] => 1
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,1,7,6,5,4,3] => 1
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6,7] => 1
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,3,2,7,6,5] => 2
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => 3
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,3,2,5,7,6] => 2
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,3,2,5,6,7] => 2
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,6,5,4] => 2
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,5,4,7] => 3
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6] => 3
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => 3
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,6,5] => 2
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => 3
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,2,4,5,7,6] => 2
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => 2
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 1
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,5,4,3,7] => 2
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => 2
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => 2
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => 2
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => 3
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => 2
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => 2
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => 1
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,3,6,5,4,7] => 2
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => 2
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => 2
111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => 1
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => 2
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 1
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [8,7,6,5,4,3,2,1] => 0
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7,6,5,4,3,2,1,8] => 1
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Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
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..
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
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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