Identifier
Values
0 => [1] => [1,0] => [1,0] => 0
1 => [1] => [1,0] => [1,0] => 0
00 => [2] => [1,1,0,0] => [1,0,1,0] => 1
01 => [1,1] => [1,0,1,0] => [1,1,0,0] => 0
10 => [1,1] => [1,0,1,0] => [1,1,0,0] => 0
11 => [2] => [1,1,0,0] => [1,0,1,0] => 1
001 => [2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
010 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
011 => [1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
100 => [1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
101 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
110 => [2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 3
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 3
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 4
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 4
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5
1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5
1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
Map
bounce path
Description
The bounce path determined by an integer composition.
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 $(\gamma^{(-1)}\circ\beta)(D)$.