Identifier
Values
0 => [2] => 1
1 => [1,1] => 2
00 => [3] => 1
01 => [2,1] => 2
10 => [1,2] => 2
11 => [1,1,1] => 3
000 => [4] => 1
001 => [3,1] => 2
010 => [2,2] => 2
011 => [2,1,1] => 3
100 => [1,3] => 2
101 => [1,2,1] => 2
110 => [1,1,2] => 3
111 => [1,1,1,1] => 4
0000 => [5] => 1
0001 => [4,1] => 2
0010 => [3,2] => 2
0011 => [3,1,1] => 3
0100 => [2,3] => 2
0101 => [2,2,1] => 2
0110 => [2,1,2] => 3
0111 => [2,1,1,1] => 4
1000 => [1,4] => 2
1001 => [1,3,1] => 2
1010 => [1,2,2] => 2
1011 => [1,2,1,1] => 3
1100 => [1,1,3] => 3
1101 => [1,1,2,1] => 3
1110 => [1,1,1,2] => 4
1111 => [1,1,1,1,1] => 5
00000 => [6] => 1
00001 => [5,1] => 2
00010 => [4,2] => 2
00011 => [4,1,1] => 3
00100 => [3,3] => 2
00101 => [3,2,1] => 2
00110 => [3,1,2] => 3
00111 => [3,1,1,1] => 4
01000 => [2,4] => 2
01001 => [2,3,1] => 2
01010 => [2,2,2] => 2
01011 => [2,2,1,1] => 3
01100 => [2,1,3] => 3
01101 => [2,1,2,1] => 3
01110 => [2,1,1,2] => 4
01111 => [2,1,1,1,1] => 5
10000 => [1,5] => 2
10001 => [1,4,1] => 2
10010 => [1,3,2] => 2
10011 => [1,3,1,1] => 3
10100 => [1,2,3] => 2
10101 => [1,2,2,1] => 2
10110 => [1,2,1,2] => 3
10111 => [1,2,1,1,1] => 4
11000 => [1,1,4] => 3
11001 => [1,1,3,1] => 3
11010 => [1,1,2,2] => 3
11011 => [1,1,2,1,1] => 3
11100 => [1,1,1,3] => 4
11101 => [1,1,1,2,1] => 4
11110 => [1,1,1,1,2] => 5
11111 => [1,1,1,1,1,1] => 6
=> [1] => 1
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Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
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.