Identifier
Values
0 => [2] => [1,1,0,0] => 0
1 => [1,1] => [1,0,1,0] => 1
00 => [3] => [1,1,1,0,0,0] => 0
01 => [2,1] => [1,1,0,0,1,0] => 1
10 => [1,2] => [1,0,1,1,0,0] => 1
11 => [1,1,1] => [1,0,1,0,1,0] => 1
000 => [4] => [1,1,1,1,0,0,0,0] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 3
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 2
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
=> [1] => [1,0] => 0
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Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module.
For the first 196 values, the statistic also gives the number of indecomposable non-projective modules $X$ such that $\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.
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.