Opened 10 years ago

Closed 7 years ago

# CycleIndexSeries derivative, integral, exponential methods are not combinatorial

Reported by: Owned by: Andrew Gainer-Dewar Andrew Gainer-Dewar major sage-6.8 combinatorics LazyPowerSeries, species Andrew Gainer-Dewar Martin Rubey N/A documentation ab9c7d1 ab9c7d161b8318bc5edffb3bb3dd66bc5c503318

The CycleIndexSeries? class inherits derivative, integral, and exponential methods from its parent class LazyPowerSeries?. However, these operations (which are perfectly reasonable at the LazyPowerSeries? level) are not combinatorially natural. In fact, there is a differentiation operator on cycle index series, but it's something entirely different, and anti-differentiation is in general impossible!

In addition to being confusing, this discrepancy has the potential to hinder work using the CycleIndexSeries? class. The combinatorial operation of cycle index differentiation is very useful in enumerative work.

All that said, the existing exponential and derivative methods are used for algebraic magic tricks in some of the existing code, so it's important not to take it away altogether.

The attached patch makes the following changes:

• Adds _lps_derivative, _lps_integral, and _lps_exponential methods to CycleIndexSeries? which call up using super.
• Refactors the internal code which previously used exponential and derivative to use _lps_exponential and _lps_derivative instead.
• Implements a new derivative method which computes the combinatorial derivative of a cycle index series.
• Implements a new pointing method which implements a related combinatorial operation.
• Implements a new integral method which raises a NotImplementedError? to prevent any user confusion (since, as explained in the docstring, there may be infinitely many very distinct cycle indices with a given derivative).
• Implements a new exponential method which returns the composition E(self) for E the cycle index series of SetSpecies?.
• Implements a new logarithm method which returns the composition Ω(self) for Ω the cycle index series implemented in CombinatorialLogarithmSeries?.

Docstrings and doctests are included for everything but the _lps_* methods, which just call up to super. I'm not sure what a doctest for these would even look like, but I'm open to suggestions.

### comment:1 Changed 10 years ago by Andrew Gainer-Dewar

For the patchbot:

apply trac_14846_cis_deriv_int_exp_methods.patch

### comment:2 Changed 10 years ago by Andrew Gainer-Dewar

Status: new → needs_review

### comment:3 Changed 10 years ago by Andrew Gainer-Dewar

I have uploaded a new version of the patch which makes a small change: rather than building the terms of the derivative of a CycleIndexSeries? using an opaque list concatenation, I have used the CycleIndexSeriesRing?.term method.

### comment:4 Changed 10 years ago by Andrew Gainer-Dewar

Branch: → u/agd/cis_deriv_int_exp_methods

### comment:5 Changed 9 years ago by Jeroen Demeyer

Please make it clear whether the patch or the git branch should be merged. In the latter case, change the milestone to sage-6.0.

### comment:6 Changed 9 years ago by Andrew Gainer-Dewar

Branch: u/agd/cis_deriv_int_exp_methods

### comment:7 Changed 9 years ago by Andrew Gainer-Dewar

Description: modified (diff)

### comment:8 Changed 9 years ago by Andrew Gainer-Dewar

Authors: → Andrew Gainer-Dewar

Rebased patch to apply over 5.12.beta4.

### comment:9 Changed 9 years ago by Frédéric Chapoton

Status: needs_review → needs_work → documentation

### comment:10 Changed 9 years ago by Andrew Gainer-Dewar

Branch: → u/agd/cis/deriv → 5ce33898e941771132cdf94837a4baf16a2cfddb sage-5.13 → sage-6.0 needs_work → needs_review

Skeletal doctests added for the three _lps_* functions. I have no idea whether this is the *right* way to test this sort of method, but at least it's *a* test.

Additionally, I've switched back over to using Git to manage this—the old Mercurial workflow is just too confusing. How do I tell the build bot to ignore the attachments and only use the branch?

### comment:11 Changed 9 years ago by For batch modifications

Milestone: sage-6.0 → sage-6.1

### comment:12 Changed 9 years ago by git

Commit: 5ce33898e941771132cdf94837a4baf16a2cfddb → 5547d749d9136f0eddf4400237183b78245e3f51

Branch pushed to git repo; I updated commit sha1. New commits:

 ​5547d74 `Merge branch 'develop' into cis/deriv`

### comment:13 Changed 9 years ago by Martin Rubey

Reviewers: → mantepse

### comment:14 follow-up:  16 Changed 9 years ago by Martin Rubey

Hi Andrew,

I'd suggest to really remove access to the three inherited methods, and rewrite the methods that use _lps_exponential accordingly:

• in set_species, we really want to return the "true" exponential cycle index series
• in partition_species, the "true" exponential of the "true" exponential - 1
• in subset_species, the square of the "true" exponential

Finally, the implementation of the "true" exponential can be taken from what's in set_species in sage 6.0.

As far as I can see, the other two methods _lps_derivative and _lps_integral are not used anywhere.

This might have a speed penalty, but I'd worry about that only if it's serious. I think it's not a good idea to make the cycle index series depend on the species code, as in the proposed patch.

I would perhaps also put the definition for the combinatorial logarithm into the same file. I think at some point we should reorganise the files in the species directory..

Martin

Last edited 9 years ago by Martin Rubey (previous) (diff)

### comment:15 Changed 9 years ago by git

Branch pushed to git repo; I updated commit sha1. New commits:

 ​dd77e4a `Move log and exp series code to generating_series.py` ​ca21d87 `Redesign patch to eliminate _lps_* methods and bad dependencies` ​c1998f2 `Merge remote-tracking branch 'origin/develop' into cis/deriv`

### comment:16 in reply to:  14 Changed 9 years ago by Andrew Gainer-Dewar

Thanks for the suggestions!

I'd suggest to really remove access to the three inherited methods, and rewrite the methods that use _lps_exponential accordingly:

Sounds good to me. Done, in the latest push.

• in set_species, we really want to return the "true" exponential cycle index series
• in partition_species, the "true" exponential of the "true" exponential - 1
• in subset_species, the square of the "true" exponential

I have rewritten all of these to use algebraic operations on the exponential CIS.

As far as I can see, the other two methods _lps_derivative and _lps_integral are not used anywhere.

Seems to be true. I just included them for completeness. They're gone now.

This might have a speed penalty, but I'd worry about that only if it's serious. I think it's not a good idea to make the cycle index series depend on the species code, as in the proposed patch.

Actually, I think the new way should actually be *faster*, there's just one instance of the exponential series which gets cached and handed around.

I would perhaps also put the definition for the combinatorial logarithm into the same file.

Once I wrote the other stuff, this seemed very reasonable, so I've done that as well.

### comment:17 Changed 9 years ago by Martin Rubey

Status: needs_review → positive_review

### comment:18 Changed 9 years ago by Volker Braun

Reviewer name must be full name. Also please fill in your name on the trac.sagemath.org homepage.

### comment:19 Changed 9 years ago by Martin Rubey

Reviewers: mantepse → Martin Rubey

### comment:21 follow-up:  25 Changed 9 years ago by Mike Hansen

Status: positive_review → needs_info

One (minor) issue is the removal of combinatorial_logarithm.py which might need to be deprecated. I've also done a "rebase" of this on top of #15673 which makes the code a little bit cleaner. This is in branch "u/mhansen/ticket/14846" . Ideally, I'd prefer to have #15673 go in first since the rebase/merge of this ticket is easier than the other way around.

Last edited 9 years ago by Mike Hansen (previous) (diff)

### comment:22 Changed 9 years ago by For batch modifications

Milestone: sage-6.1 → sage-6.2

### comment:23 Changed 9 years ago by git

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

 ​30fcfde `Initial work on new streams` ​96e4ea3 `Make streams know about their order / aorder` ​a076622 `Move generating_series over to new streams format` ​9e56173 `Fix issue in LazyPowerSeries.__repr__ with (eventually) constant streams` ​fa83ccc `More work on moving generating_series over to new format` ​3b56620 `Fix repr in recursive species` ​ea1902a `Fix cycle species cis` ​458bdaf `Fix bug in generating_series` ​9b038bb `Clean up order_operation and make the default to return 0` ​fefaacf `Start making the basic species use Streams directly`

### comment:24 Changed 9 years ago by git

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

 ​a630924 `Implement combinatorial derivative, etc. for CycleIndexSeries` ​5ce3389 `Add doctests for _lps_* methods` ​5547d74 `Merge branch 'develop' into cis/deriv` ​c1998f2 `Merge remote-tracking branch 'origin/develop' into cis/deriv` ​ca21d87 `Redesign patch to eliminate _lps_* methods and bad dependencies` ​dd77e4a `Move log and exp series code to generating_series.py` ​6989f3f `Merge remote-tracking branch 'origin/develop' into cis-deriv` ​2d3b466 `Merge remote-tracking branch 'origin/u/mhansen/ticket/14846' into cis-deriv` ​4efe9b0 `Rename CycleIndexSeriesRing.omega() to logarithm_series` ​334c095 `Rename CycleIndexSeriesRing().exponential() to exponential_series()`

### comment:25 in reply to:  21 Changed 9 years ago by Andrew Gainer-Dewar

Dependencies: → 15673

One (minor) issue is the removal of combinatorial_logarithm.py which might need to be deprecated.

Ah, yes, this is an important point. I sometimes forget that other people might have used my code… =D

I've also done a "rebase" of this on top of #15673 which makes the code a little bit cleaner. This is in branch "u/mhansen/ticket/14846" . Ideally, I'd prefer to have #15673 go in first since the rebase/merge of this ticket is easier than the other way around.

I don't yet understand everything that's happening with #15673, but I'm starting to take a look at it. I'm definitely open to the idea, though—better series code will make all our work easier! I've taken a look at your rebased version of this, and have a few thoughts up-front:

• In the docstring for the deprecated `CombinatorialLogarithmSeries()`, you point the reader to `CycleIndexSeriesRing(R).exponential()`, but in fact this should be to `CycleIndexSeriesRing(R).omega()`.
• Relatedly, I'd argue that the `CycleIndexSeriesRing(R).omega()` and `CycleIndexSeriesRing(R).exponential()` methods should instead be named `logarithm_series()` and `exponential_series()`. In the first place, the Ω notation is not totally standard (it's used by Labelle, but other authors have called this virtual species "Con" or other things); in the second, I think it's conceptually important to emphasize that these objects are series and not operations, since the operations are implemented as `CycleIndexSeries().exponential()` and `CycleIndexSeries().logarithm()` respectively. (Of course, the documentation should mention the relationship between the two!)
• `CycleIndexSeriesRing.LogarithmStream` appears to be internal, so the docstring of `CycleIndexSeries.logarithm()` should refer instead to `CycleIndexSeriesRing.logarithm_series`.

I've updated the ticket branch to use your code with these changes. I've also set a dependency on #15673. (Evidently, I bungled the ticket modifications a bit, but I think it's all sorted now.)

New commits:

 ​6989f3f `Merge remote-tracking branch 'origin/develop' into cis-deriv` ​2d3b466 `Merge remote-tracking branch 'origin/u/mhansen/ticket/14846' into cis-deriv` ​4efe9b0 `Rename CycleIndexSeriesRing.omega() to logarithm_series` ​334c095 `Rename CycleIndexSeriesRing().exponential() to exponential_series()`
Last edited 9 years ago by Andrew Gainer-Dewar (previous) (diff)

### comment:26 Changed 9 years ago by Martin Rubey

Dependencies: 15673 → #15673

### comment:27 Changed 9 years ago by git

Commit: 334c0956250038eae2d6efc915231e650b13be79 → 538dde76d0e116f67e1115337193fe12c48ce62b

Branch pushed to git repo; I updated commit sha1. New commits:

 ​538dde7 `Fix broken docstring in combinatorial_logarithm.py`

### comment:28 Changed 9 years ago by Mike Hansen

Cool, your changes look good to me!

### comment:29 Changed 9 years ago by For batch modifications

Milestone: sage-6.2 → sage-6.3

### comment:30 follow-up:  32 Changed 9 years ago by Andrew Gainer-Dewar

Dependencies: #15673

While #15673 looks like it will bring some much-needed improvements to the algebraic machinery of species in Sage, it also looks like it's going to take a while to implement. I've been fielding questions from other researchers who would like to use the code in #14347, which definitely depends on this one, so I'd be very grateful if we could move forward on this one now. I will, of course, be happy to revisit the issue in the context of #15673 once that situation stabilizes!

### comment:31 Changed 8 years ago by For batch modifications

Milestone: sage-6.3 → sage-6.4

### comment:32 in reply to:  30 Changed 8 years ago by Martin Rubey

I agree. I wonder what happened to #16137.

### comment:33 Changed 8 years ago by Frédéric Chapoton

Milestone: sage-6.4 → sage-6.6

### comment:34 Changed 8 years ago by git

Commit: 538dde76d0e116f67e1115337193fe12c48ce62b → ab9c7d161b8318bc5edffb3bb3dd66bc5c503318

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

 ​ab9c7d1 `Rebase #14846 on current mainline Sage`

### comment:35 Changed 8 years ago by Andrew Gainer-Dewar

Since #15673 seems to have died on the vine, I have rebuilt the code for this commit on top of current mainline Sage 6.7. All doctests in `combinat/species` pass.

### comment:36 Changed 8 years ago by Andrew Gainer-Dewar

Milestone: sage-6.6 → sage-6.8 needs_info → needs_review

### comment:37 Changed 7 years ago by Martin Rubey

Status: needs_review → positive_review

I played with the code and the examples, and found myself happy. For example,

```sage: T = CombinatorialSpecies()
sage: X = species.SingletonSpecies()
sage: E = species.SetSpecies()
sage: T.define(X*E(T))
sage: s = T.cycle_index_series()
sage: oeis(s.logarithm().generating_series().counts(12))
0: A133297: a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*n^(n-k-1)/(n-k)!.
sage: oeis(s.pointing().generating_series().counts(12)[1:])
0: A000312: Number of labeled mappings from n points to themselves (endofunctions): n^n.
1: A177885: (1-n)^(n-1).
2: A086783: Discriminant of the polynomial x^n - 1.
sage: oeis(s.pointing().isotype_generating_series().counts(12)[1:])
0: A000107: Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.
```

Thanks for the code and thanks for the patience!

### comment:38 Changed 7 years ago by Volker Braun

Branch: u/agd/cis/deriv → ab9c7d161b8318bc5edffb3bb3dd66bc5c503318 → fixed positive_review → closed
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