#14543: Implement compositional inverses of cycle index series
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Reporter: agd | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.11
Component: combinatorics | Resolution:
Keywords: species, cycle indices | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by agd):
Replying to [comment:9 darij]:
> Ooh -- this is not literally the compositional inverse of a power
series? Sorry then; please disregard what I wrote. It would still be nicer
to have some explanation of what exactly this is, or a more precise
reference (to a section number?).
Formally, this is the inverse of the operation of "cycle index plethysm",
which is handled by the `composition` method of `CycleIndexSeries`. As an
abstract operation, it has many of the same properties as power series
composition, but at the computational level it is somewhat different; in
particular, the induced operation on coefficients is something a bit more
subtle than simply multiplying things out.
That said, the inversion procedure as I've written it out doesn't actually
depend on any of the details of the operation ∘. It suffices that there is
a two-sided ∘-inverse (call it X) and that ∘ distributes over addition.
Thus, writing this procedure up at the `LazyPowerSeries` level and letting
`CycleIndexSeries` inherit it should work fine, since `CycleIndexSeries`
appropriately overrides `composition`.
The standard reference on the subject is section 1.4 of "Combinatorial
species and tree-like structures" by Bergeron, Labelle, and Leroux,
although it is discussed at some length in many other sources as well.
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14543#comment:10>
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