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

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 product of Schubert classes in the cohomology ring of the Grassmannian

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

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