Identifier
-
Mp00275:
Graphs
—to edge-partition of connected components⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001028: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(1,2)],3) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(0,2),(1,2)],3) => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 3
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(2,3)],4) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(1,3),(2,3)],4) => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 3
([(0,3),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 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] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(3,4)],5) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(2,4),(3,4)],5) => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 3
([(1,4),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
([(1,4),(2,3),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,1),(2,4),(3,4)],5) => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 2
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 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] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 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] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(4,5)],6) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(3,5),(4,5)],6) => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 3
([(2,5),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
([(2,5),(3,4),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(1,2),(3,5),(4,5)],6) => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 2
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 4
([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 3
([(1,5),(2,4),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 4
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 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] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 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] => [1,1,0,1,0,1,0,1,1,0,0,0] => 4
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(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] => [1,1,0,1,1,0,0,0,1,0] => 4
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0] => 3
([(5,6)],7) => [1] => [1,0,1,0] => [1,1,0,0] => 2
([(4,6),(5,6)],7) => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 3
([(3,6),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
([(2,6),(3,6),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 3
([(3,6),(4,5)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
([(3,6),(4,5),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 3
>>> Load all 185 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
Delest-Viennot-inverse
Description
Return the Dyck path obtained by applying the inverse of Delest-Viennot's bijection to the corresponding parallelogram polyomino.
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 $(\beta^{(-1)}\circ\gamma)(D)$.
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 $(\beta^{(-1)}\circ\gamma)(D)$.
Map
to edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.
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