# LRnumber -- returns the number of solutions to the given Schubert problem

## Synopsis

• Usage:
LRnumber(conditions,k,n)
• Inputs:
• conditions, a list, of Schubert conditions, either partitions or brackets, that constitutes a Schubert problem on the Grassmannian $Gr(k,n)$.
• k, an integer,
• n, an integer, $k$ and $n$ define the Grassmannian $Gr(k,n)$ of $k$-planes in $n$-space
• Optional inputs:
• Strategy => ..., default value Schubert2, strategy for computing the number of solutions to a Schubert problem
• Outputs:
• an integer, The number of solutions to the given Schubert problem

## Description

This first verifies that the conditions are either all partitions or all brackets, and that they form a Schubert problem on $Gr(k,n)$.

Then it computes the intersection number of the prodiuct of Schubert classes in the cohomology ring of the Grassmannnian

For instance, the problem of four lines is given by 4 partitions {1}$^4$ in $Gr(2,4)$

 i1 : LRnumber({{1},{1},{1},{1}},2,4) o1 = 2

the same problem but using brackets instead of partitions

 i2 : LRnumber({{2,4},{2,4},{2,4},{2,4}},2,4) o2 = 2

the same problem but using phc implementation of Littlewood-Richardson rule

 i3 : LRnumber({{1},{1},{1},{1}},2,4,Strategy => "phc") o3 = 2

## Caveat

This uses the package Schubert2 and the Strategy "phc" requires the string parsing capabilities of Macaulay2 version 1.17 or later

## See also

• LRrule -- computes the product of Schubert classes using geometric Littlewood-Richardson rule

## Ways to use LRnumber :

• "LRnumber(List,ZZ,ZZ)"

## For the programmer

The object LRnumber is .