Identifier
Values
([],1) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([],2) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,1)],2) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,1),(0,2)],3) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,2),(2,1)],3) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,2),(1,2)],3) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,1),(0,2),(1,3),(2,3)],4) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(3,1),(3,2)],4) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(1,3),(3,2)],4) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(1,2)],4) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,3),(1,2),(1,3)],4) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,3),(2,1),(3,2)],4) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(0,4),(3,2),(4,1)],5) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,4),(1,4),(2,3),(4,2)],5) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,3),(3,4),(4,1),(4,2)],5) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(2,3),(3,1),(4,2)],5) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,3),(1,2),(2,4),(3,4)],5) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1
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
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
promotion cycle type
Description
The cycle type of promotion on the linear extensions of a poset.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.