Identifier
Values
['A',1] => ([],1) => ([],1) => 1
['A',2] => ([(0,2),(1,2)],3) => ([(0,1)],2) => 2
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(2,1)],3) => 3
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
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       
Description
The number of left modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Map
maximal antichains
Description
The lattice of maximal antichains in a poset.
An antichain $A$ in a poset is maximal if there is no antichain of larger cardinality which contains all elements of $A$.
The set of maximal antichains can be ordered by setting $A \leq B \Leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow}B$, where $\mathop{\downarrow}A$ is the order ideal generated by $A$.
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.