#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:8 nthiery]:
> For the record: I recently needed the compositional inverse for Lazy
power series, and a couple other related operations; they are in the patch
I am about to attach; you can also find it in the queue. Feel free to take
over / recycle whatever might be relevant.
Unfortunately, I don't think this approach will work at the
`CycleIndexSeries` level. The `compose_inverse_with` method from your
patch, it appears that you're using the Inverse Function Theorem. This
actually ''does'' work at the level of cycle index series (by a sort of
combinatorial chain rule!), but the resulting differential equation is
intractable in general, because taking species-theoretic derivatives of
cycle index series destroys ''lots'' of information (in particular, every
term without a p_1 is destroyed).
I haven't dug too deeply into the algebra, but I am fairly confident that
using `LazyPowerSeries` derivatives instead is not mathematically
meaningful and won't yield anything useful.
Looking this over has, however, brought to my attention that Sage does not
have a method for `CycleIndexSeries` implementing the ''species-
theoretic'' derivative. Curiously, the `LazyPowerSeries` `derivative()`
and `integral()` methods are used several times in the existing code-base
to pull off algebra tricks, so factoring out these (in my opinion)
deceptive names would be a bit of a hassle, but it might still be
worthwhile. This is probably at least as much a cultural question as a
code one, though, so perhaps I should steer clear until I get to know the
environment better?
Also, since the patchbot now seems to be confused:
apply trac_14543_cycle_index_compositional_inverse.patch
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14543#comment:11>
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