Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(1,2)],3) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(0,2),(1,2)],3) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 2
([(2,3)],4) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(1,3),(2,3)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 2
([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
([(3,4)],5) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(2,4),(3,4)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 2
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
([(4,5)],6) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(3,5),(4,5)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 2
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 3
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 4
([(5,6)],7) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([(4,6),(5,6)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
>>> Load all 262 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
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 $1 \tilde D_2 0 \tilde D_1$.
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 $1 \tilde D_2 0 \tilde D_1$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to edge-partition of biconnected components
Description
Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.
The biconnected components are also known as blocks of a graph.
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