Identifier
Values
0 => [2] => [1,1,0,0] => [2,3,1] => 1
1 => [1,1] => [1,0,1,0] => [3,1,2] => 1
00 => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 1
01 => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
10 => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
11 => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 5
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
100 => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 5
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 4
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 11
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 9
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 6
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 9
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 5
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => 14
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => 10
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => 5
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 10
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => 14
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 5
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 1
=> [1] => [1,0] => [2,1] => 1
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Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
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.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.