Profile of a Finite Permutation Group

[falque]

Add methods profile and profile_series (= profile_polynomial) in category finite_permutation_groups, in order to be able to compute the orbital profile, and its generating series/polynomial, of a finite permutation group.

(The profile of G maps any non negative integer n onto the number of G-orbits of n-subsets, for the induced action of G on all subsets of its domain.)

CC: @nthiery @nadialafreniere

Component: group theory

Keywords: permutation groups, profile, fpsac2019

Author: Justine Falque

Branch/Commit: 609ea59

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/28172

[falque]

Branch: u/falque/profile_of_a_finite_permutation_group

[fchapoton]

Commit: fde25b1

[fchapoton]

comment:2

after EXAMPLES:: the doctests must be indented by 4 spaces

in "profile", add INPUT to explain whar is n

add doctests with another variable name

add doc and doctests for "profile_series"

---

New commits:

e9b8464	added profile and polynomial_of_profile methods to class PermutationGroup_generic
fde25b1	28172 added methods profile and profile_polynomial to class PermutationGroup_generic

[fchapoton]

comment:3

if profile_series is just an alias, you can write

profile_series = profile_polynomial

[nthiery]

comment:4

Thanks Frédéric for the review! I also gave some oral feedback to Justine.

if profile_series is just an alias, you can write

profile_series = profile_polynomial

Yup, with the caveat that if a subclass overrides profile_polynomial, profile_series will still point to the one here.

[falque]

Changed branch from u/falque/profile_of_a_finite_permutation_group to none

[falque]

Changed commit from fde25b1 to none

[falque]

comment:5

Methods should be added to the category of finite permutation groups rather than the class PermutationGroup_generic

[falque]

Branch: u/falque/profile_of_a_finite_permutation_group

[sagetrac-git]

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

6ede59a

Improvements in the documentation and more flexibility about the nature of variable

[sagetrac-git]

Commit: 6ede59a

[nadialafreniere]

Changed keywords from permutation groups, profile to permutation groups, profile, fpsac2019

[fchapoton]

comment:9

needs review ?

[nadialafreniere]

comment:12

Great job Justine! I'm glad you added it to Sage.
I have some comments for you:

• It should be clear in the doc what the profile is. Please, add it to the docstring of profile_polynomial.
• There could be more examples in profile_polynomial. For now, the only example is with the fifth cyclic group, but it would be nice to have it with something else. For example, C6 would be interesting, since it makes it clear that we are looking only at cycles and no other symmetries, like with D6 (it was not exactly clear to me after the computation for C5, since the profile polynomial is the same as for D5). The Symmetric or Dihedral groups could be good examples.
• It is not clear to me what the added value of the profile_series function is. Doesn't it do the same as profile_polynomial? If so, I like the idea of an alias. If you are worried about someone changing profile_polynomial, it is still an issue in the way it is written right now.
• I don't know how to do it efficiently, but I am wondering if it really is a good idea to compute the whole polynomial just to get one coefficient in profile. If there is a faster way to do it, especially when the group is big, please change the code.
• There are some trailing spaces at the end of lines, and they should be erased.
  Congrats againt for this contribution!

[sagetrac-git]

Changed commit from 6ede59a to 012740b

[sagetrac-git]

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

012740b

Revised docstring, particularly in profile_series, with new examples and a description of what the profile is; profile_polynomial becomes an alias; an optional argument is added in method profile to allow the user to change the computation method.

[sagetrac-git]

Changed commit from 012740b to fd6fbb4

[sagetrac-git]

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

fd6fbb4

wrong file, sorry; corrected

[falque]

comment:15

• I made the suggested additions to the documentation of profile_series.

• "If you are worried about someone changing profile_polynomial, it is still an issue in the way it is written right now."
  --> I was told differently indeed; anyway, I think it is clearer with
  an alias for now, so I changed it.

• I checked the useless spaces, it should be ok.

• "I don't know how to do it efficiently, but I am wondering if it really
  is a good idea to compute the whole polynomial just to get one coefficient
  in profile. If there is a faster way to do it, especially when the group
  is big, please change the code."
  --> There is indeed a more direct method, although I would call it the
  "naive" method since I am not sure it is really faster than Polyà
  enumeration, which presents in addition the bonus of computing all
  values of the profile in a row. I added nevertheless an optional argument
  to allow the user to use their preferred method. This can also make what
  the profile is clearer to the user !

• some micro changes here and there

[fchapoton]

comment:16

• Latex syntax is not allowed in the doc, but unicode works fine : instead of P'{o}ly
`{a}, just use Pólya.

• You could add some cross-references using
  .. SEEALSO:: :meth:`profile` and
  .. SEEALSO:: :meth:`profile_series`
  in the appropriate place of the docs.

[sagetrac-git]

Changed commit from fd6fbb4 to 798f9c0

[sagetrac-git]

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

798f9c0

Added references, fixed one or two typos and errors

[fchapoton]

comment:18

Pólya takes just one accent:

https://fr.wikipedia.org/wiki/George_P%C3%B3lya

[sagetrac-git]

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

609ea59

typo

[sagetrac-git]

Changed commit from 798f9c0 to 609ea59

[fchapoton]

comment:20

ok, looks good to me. Merci !

[fchapoton]

Reviewer: Frédéric Chapoton

[falque]

comment:21

Replying to @fchapoton:

ok, looks good to me. Merci !

Thank you !!!

[vbraun]

Changed branch from u/falque/profile_of_a_finite_permutation_group to 609ea59