Safe Haskell | None |
---|---|
Language | Haskell2010 |
Geometry of the degree n!
finite map pi
, which just forgets the order points:
pi : Q^n = P^1 x P^1 x ... x P^1 -> P^n = P(Sym^n C^2)
It's clear that the degree of pi
restricted to an open stratum corresponding to
a partition mu
is the multinomial coefficient corresponding to n
.choose
mu
It is also not hard to see that the degree of pi
restricted to the intersection
of the open stratum corresponding to mu
with the image of the diagonal map
corresponding to nu
equals the number of "automorphisms" aut(mu) = prod (e_i!)
where mu = (1^e1 2^e2 ... k^ek)
and the number of ways nu
is refinement of mu
.
Note that for nu=(1,1...1)
the multinomial agrees with the number of refinements.
This module contains functions to compute these numbers.
Synopsis
- countCoarsenings :: Partition -> Map Partition Integer
- countDirectCoarsenings :: Partition -> Map Partition Integer
- countCoarseningsNaive :: Partition -> Map Partition Integer
- countPreimage :: Partition -> Map Partition Integer
- preimageView :: Partition -> (Integer, Map Partition Integer)
- countFullPreimage :: Partition -> Integer
Documentation
countCoarsenings :: Partition -> Map Partition Integer Source #
Given a partition, we list all coarser partitions together with the number of ways the input is a refinement of the coarser partition.
TODO: at the moment this is just a synonym for countCoarseningsNaive
...
countDirectCoarsenings :: Partition -> Map Partition Integer Source #
Count coarsenings (with multiplicities) which are shorter by just 1.
countCoarseningsNaive :: Partition -> Map Partition Integer Source #
Naive (very slow) implementation of countCoarsenings
.
countPreimage :: Partition -> Map Partition Integer Source #
Given a partition nu
, we stratify the image of the
corresponding diagonal Delta_nu
as usual, and list
the degree of pi
restricted to these strata
This is just counting the coarsenings, multiplied by the number of "automorphisms" of the partition.
preimageView :: Partition -> (Integer, Map Partition Integer) Source #
The preimage counts, but the partition itself is separated out.
countFullPreimage :: Partition -> Integer Source #
The preimage pi^-1(x)
of a point under the map
pi : Q^n -> P^n
is just a multinomial coefficient