Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000638: Permutations ⟶ ℤ
Values
0 => [1] => [1,0] => [1] => 1
1 => [1] => [1,0] => [1] => 1
00 => [2] => [1,1,0,0] => [1,2] => 1
01 => [1,1] => [1,0,1,0] => [2,1] => 2
10 => [1,1] => [1,0,1,0] => [2,1] => 2
11 => [2] => [1,1,0,0] => [1,2] => 1
000 => [3] => [1,1,1,0,0,0] => [1,2,3] => 1
001 => [2,1] => [1,1,0,0,1,0] => [3,1,2] => 3
010 => [1,1,1] => [1,0,1,0,1,0] => [3,2,1] => 2
011 => [1,2] => [1,0,1,1,0,0] => [2,3,1] => 2
100 => [1,2] => [1,0,1,1,0,0] => [2,3,1] => 2
101 => [1,1,1] => [1,0,1,0,1,0] => [3,2,1] => 2
110 => [2,1] => [1,1,0,0,1,0] => [3,1,2] => 3
111 => [3] => [1,1,1,0,0,0] => [1,2,3] => 1
0000 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 3
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 3
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 3
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 4
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 4
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 3
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 3
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 3
1111 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 3
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 3
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 3
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 3
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 3
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 5
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 3
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 4
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 4
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 4
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 4
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 4
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 4
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 4
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 2
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 4
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 2
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 5
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 3
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 3
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 3
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 3
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 1
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 3
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 3
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 3
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 3
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => 3
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 5
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 3
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => 3
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => 5
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => 3
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => 3
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => 5
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => 5
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => 5
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 3
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 4
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 4
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 4
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 2
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 4
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 2
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 4
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 6
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 4
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 4
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 4
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 4
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 4
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 2
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 2
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 4
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 4
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 4
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 4
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 4
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 6
>>> Load all 126 entries. <<<
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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
$643$.
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
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
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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