- FindStat

Identifier

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000384: Integer partitions ⟶ ℤ

Values

['A',1] => ([],1) => [2] => 2
['A',2] => ([(0,2),(1,2)],3) => [3,2] => 3
['B',2] => ([(0,3),(1,3),(3,2)],4) => [4,2] => 4
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => 6

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

click to show known generating functions

$F_{1} = q^{2}$

$F_{2} = q^{3} + q^{4} + q^{6}$

Description

The maximal part of the shifted composition of an integer partition.
A partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding

$i-1$ to the

$i$ -th part.
The statistic is then

$\operatorname{max}_i\{ \lambda_i + i - 1 \}$ .
See also St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition..

Map

rowmotion cycle type

Description

The cycle type of rowmotion on the order ideals of a poset.

Map

to root poset

Description

The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where

$\alpha \prec \beta$ if

$\beta - \alpha$ is a simple root.