Cardinality of partitions of negative values

[tscrim]

This should work (as long as it accepts negative values, which there is nothing wrong with it, it is just an empty set), but it results in an overflow error:

sage: Partitions(-5)
Partitions of the integer -5
sage: _.cardinality()

CC: @darijgr @fchapoton @jhpalmieri

Component: combinatorics

Author: Travis Scrimshaw

Branch/Commit: c340478

Reviewer: John Palmieri

Issue created by migration from https://trac.sagemath.org/ticket/33323

[tscrim]

Commit: c340478

[tscrim]

comment:1

That is for the default using flint, but gap is wrong:

sage: Pm5.cardinality(algorithm='gap')
1
sage: Pm5.cardinality(algorithm='pari')
0

This is needed for doctests on #30680.

---

New commits:

c340478

Partitions of negative n is always 0.

[tscrim]

Branch: public/combinat/negative_partition_card-33323

[fchapoton]

comment:2

I would rather think that most combinatorial objects return a ValueError:
see DyckWords(-2), BinaryTrees(-2), Posets(-2), Tableaux(-2) and TamariIntervalPosets(-2).

AlternatingSignMatrices(-2) returns an ArithmeticError.

But Compositions(-2), SuperPartitions(-2) and Permutations(-2)"work".

If you prefer, returning the empty set would be another solution. Do we have such an object ?

[tscrim]

comment:3

Perhaps it would be better for these to raise a ValueError to be consistent with other objects. However, we would also need to make sure that this continues to work:

sage: s = SymmetricFunctions(QQ).s()
sage: list(s.basis(-2))
[]

We do not seem to have a singular EmptySet object, just EmptySetError.

[tscrim]

comment:4

In order to keep that continuing to work, I think we would need a more long-term fix, which would include specifying the valid indices for the grading monoid of any graded module.

[jhpalmieri]

comment:5

This looks simple and these things work:

sage: P = Partitions(-3)
sage: P.cardinality()
0
sage: list(P)
[]
sage: for a in P: print(a)
sage:

I'm somewhat agnostic about whether Partitions(-3) should raise an error or return an empty set. It feels somewhat different in flavor than Posets(-3) because it doesn't make sense to ask for a poset with -3 elements, while you could ask for all partitions of the number -3 — there just aren't any. But I don't have strong feelings either way.

[darijgr]

comment:6

For the sake of (e.g.) the Euler pentagonal number theorem, it makes the most sense to return the empty set. Otherwise, the sum needs to be broken up at some nasty bounds.

[jhpalmieri]

comment:7

This can also be a stopgap ticket until a more permanent solution is found.

[tscrim]

comment:8

An interesting comparison point:

sage: Pm = Permutations(-3)
sage: [x for x in Pm]    
[[]]
sage: Pm.cardinality()
ValueError: factorial -- must be nonnegative

So there are bugs there that can be handled on another ticket.

[jhpalmieri]

comment:9

In my opinion we should merge this ticket and then think about long-term solutions elsewhere.

[tscrim]

comment:10

@fchapoton, any objections to John doing the review for this as a stopgap ticket?

[fchapoton]

comment:11

no objection whatsoever ; I have no monopoly on reviews

[tscrim]

comment:12

We value your opinion. Plus, all objections should be resolved before a positive review too.

[jhpalmieri]

comment:13

Let's merge this as is, and on other tickets continue to think about how to handle these situations.

[jhpalmieri]

Reviewer: John Palmieri

[vbraun]

Changed branch from public/combinat/negative_partition_card-33323 to c340478