Identifier
Values
0 => [2] => [1,1,0,0] => [2,3,1] => 3
1 => [1,1] => [1,0,1,0] => [3,1,2] => 3
00 => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 4
01 => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 5
10 => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 5
11 => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 4
000 => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 5
001 => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 7
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 8
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 7
100 => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 7
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 8
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 7
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 5
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 6
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 9
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 11
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 10
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 11
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 13
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 12
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 9
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 12
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 13
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 11
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 10
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 11
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 9
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 6
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 7
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 11
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] => 13
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] => 11
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 13
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] => 11
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] => 7
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 8
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => 8
=> [1] => [1,0] => [2,1] => 2
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Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
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.