Identifier
Values
0 => [2] => [1,1,0,0] => [1,0,1,0] => 0
1 => [1,1] => [1,0,1,0] => [1,1,0,0] => 0
00 => [3] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
01 => [2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 0
10 => [1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 0
11 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 1
000 => [4] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 3
=> [1] => [1,0] => [1,0] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.
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.