- FindStat

Identifier

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001101: Integer partitions ⟶ ℤ

Values

([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [2] => 0
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,1] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [2] => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [3] => 0
([(1,6),(2,6),(3,5),(4,5)],7) => [2,2] => [2] => 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [3,2] => [2] => 0
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [1,1] => 1
([(0,3),(1,2),(4,6),(5,6)],7) => [2,1,1] => [1,1] => 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [3,2] => [2] => 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,2] => [2] => 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,2] => [2] => 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,3] => [3] => 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,2] => [2] => 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => [5,2] => [2] => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,1] => 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [3] => 0
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [3] => 0
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [3] => 0
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => [4,3] => [3] => 0
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,2] => [2] => 0
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [3] => 0
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,3] => [3] => 0

search for individual values

searching the database for the individual values of this statistic

Description

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees.
For a generating function

$f$ the associated formal group law is the symmetric function

$f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$ , see [1].
This statistic records the coefficient of the monomial symmetric function

$m_\lambda$ times the product of the factorials of the parts of

$\lambda$ in the formal group law for increasing trees, whose generating function is

$f(x) = -\log(1-x)$ , see [1, sec. 9.1]
Fix a coloring of

$\{1,2, \ldots, n\}$ so that

$\lambda_i$ are colored with the

$i$ th color. This statistic gives the number of increasing trees on this colored set of vertices so that no leaf has the same color as its parent. (An increasing tree is a rooted tree on the vertex set

$\{1,2, \ldots, n\}$ with the property that any child of

$i$ is greater than

$i$ .)

Map

to edge-partition of connected components

Description

Sends a graph to the partition recording the number of edges in its connected components.

Map

first row removal

Description

Removes the first entry of an integer partition