Identifier
Values
0 => [1] => [1,0] => [1] => 0
1 => [1] => [1,0] => [1] => 0
00 => [2] => [1,1,0,0] => [2,1] => 0
01 => [1,1] => [1,0,1,0] => [1,2] => 0
10 => [1,1] => [1,0,1,0] => [1,2] => 0
11 => [2] => [1,1,0,0] => [2,1] => 0
000 => [3] => [1,1,1,0,0,0] => [3,2,1] => 1
001 => [2,1] => [1,1,0,0,1,0] => [2,1,3] => 0
010 => [1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 0
011 => [1,2] => [1,0,1,1,0,0] => [1,3,2] => 0
100 => [1,2] => [1,0,1,1,0,0] => [1,3,2] => 0
101 => [1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 0
110 => [2,1] => [1,1,0,0,1,0] => [2,1,3] => 0
111 => [3] => [1,1,1,0,0,0] => [3,2,1] => 1
0000 => [4] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 2
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 0
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 1
1111 => [4] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 2
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 5
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 2
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 2
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 5
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 5
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 2
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 1
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 2
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 1
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 0
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 0
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 0
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 0
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 0
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 1
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 2
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 1
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 0
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 0
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 0
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 0
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 1
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 1
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 0
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 1
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 1
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 2
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 5
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 5
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 2
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 1
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 1
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 0
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 1
>>> Load all 124 entries. <<<
101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 1
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 0
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
101011 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 0
101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 0
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 0
101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 1
101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2
110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 2
110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 1
110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 0
110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 0
110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 0
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 0
110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 0
110111 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 1
111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 2
111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 1
111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 1
111011 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 1
111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 2
111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 2
111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 5
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Description
The maximum defect over any reduced expression for a permutation and any subexpression.
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 $w$.
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..