Oscillating Tableaux

[BruceWestbury]

This implements oscillating tableaux (aka up down tableaux). These are represented as a sequence of partitions. They can also be constructed from
appropriate words or perfect matchings. I have implemented the action of
the cactus group.

Depends on #25434

CC: @kevindilks @mantepse @sagetrac-pfannerer

Component: combinatorics

Keywords: oscillating tableau, cactus group

Author: Bruce Westbury

Branch/Commit: u/Bruce/oscillating_tableaux @ 247e3d2

Reviewer: Travis Scrimshaw, Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/25557

[BruceWestbury]

Changed keywords from none to oscillating tableau, cactus group

[BruceWestbury]

Dependencies: #25434

[BruceWestbury]

Author: Bruce Westbury

[BruceWestbury]

Branch: u/Bruce/oscillating_tableaux

[sagetrac-git]

Commit: 53c8d04

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

e25e2aa	Merge branch 'develop' into t/25434/pathtableaux
6518f21	Dealt with Travis' objection to my `__getattr__` hack and most of his other comments.
ec63927	Made minor changes to catalan.py
12b27c4	Changed directory name
653f8a5	Changed import to new file names
136660e	Merge branch 't/25434/pathtableaux' into t/25557/oscillating_tableaux
6bab3e6	Made fixes so it works with new version of path_tableaux
6eb764e	Worked on getting doctests to pass
3168307	Working on doctests
53c8d04	More doctests

[sagetrac-git]

Changed commit from 53c8d04 to e596a6a

[sagetrac-git]

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

e596a6a

Edited documentation

[sagetrac-git]

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

03152b8	Moved ClonableArray from ABC class to implementation
2c77948	Merge branch 't/25434/pathtableaux' into t/25557/oscillating_tableaux
dd7f9d6	Moved ClonableArray to implementation

[sagetrac-git]

Changed commit from e596a6a to dd7f9d6

[kevindilks]

Changed branch from u/Bruce/oscillating_tableaux to u/kdilks/oscillating_tableaux

[kevindilks]

Changed commit from dd7f9d6 to bf7e8b7

[kevindilks]

comment:9

There were some minor typos in the module.rst file causing documentation build to fail.

---

New commits:

8e014c3	Merge branch 'u/Bruce/oscillating_tableaux' of git://trac.sagemath.org/sage into 25557OscTab
bf7e8b7	fixed module.rst links

[kevindilks]

comment:10

There definitely needs to be some kind of documentation string with examples for the classes themselves. Right now if I try to do OscillatingTableau? or OscillatingTableaux? in command line, instead of getting information about what inputs I should be giving, what the objects look like, what rules there are for the objects, what are some sample objects I could create to use tab completion to see what methods are available etc., I just getting information about lazy_import.

[BruceWestbury]

Changed branch from u/kdilks/oscillating_tableaux to u/Bruce/oscillating_tableaux

[sagetrac-git]

Changed commit from bf7e8b7 to 90673a6

[sagetrac-git]

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

90673a6

Added more documentation

[sagetrac-pfannerer]

comment:13

There is a minor error in OscillatingTableau.to_word:
A minimal example is

sage: OscillatingTableau([[2,1],[2,2]]).to_word()
[1]

instead of

[2]

Here is a fix:

diff --git a/src/sage/combinat/path_tableaux/oscillating.py b/src/sage/combinat/path_tableaux/oscillating.py
index b88bfb8..931553c 100644
--- a/src/sage/combinat/path_tableaux/oscillating.py
+++ b/src/sage/combinat/path_tableaux/oscillating.py
@@ -334,7 +334,7 @@ class OscillatingTableau(ClonableArray,PathTableau):
         """
         n = len(self)
         result = [0]*(n-1)
-        l = map(len,self)
+        l = map(len, self)
 
         for i in range(n-1):
             if l[i] > l[i+1]:
@@ -342,9 +342,11 @@ class OscillatingTableau(ClonableArray,PathTableau):
             elif l[i] < l[i+1]:
                 result[i] = l[i+1]
             else:
-                for u, v in zip(self[i],self[i+1]):
-                    if u != v:
-                        result[i] = v-u
+                for j in range(l[i]):
+                    d = self[i+1][j]-self[i][j]
+                    if d:
+                        result[i] = (j+1)*d
+                        break

[kevindilks]

comment:14

Next round of comments:

• documentation, Example: are things that are tested and shown to the user. Test: are things that are checked during doctests, but aren't shown to the user in documentation. So you want the code for promotion being the inverse of rotation to be in the example environment, and the various testsuites to be under a test environment.

• The convention for docstring phrasing is "Return a " instead of "Returns a ".

• Phrasing of "This has vertices words of length r in this alphabet" is a little awkward. Maybe make it "The vertices are words of length r in this alphabet".

• Would it make sense to have a possible type of input be a partition shape, along with a word indicating which rows boxes are added/removed from? It seems a little asymmetric that most forms of input allow you to start from an arbitrary shape, but word input forces you to start from the empty shape.

• Docstring for is_skew should have the fancy quotes around True and False.

• crossing_number is missing a an empty line and EXAMPLE: between the docstring text and examples.

• crossing_number and nesting number should explain that it's coming from the associated matching. It's only mentioned under the to_perfect_matching() method that there's a bijection between the oscillating tableau that start/end at the empty shape and perfect matchings, and no mention of a relationship between arbitrary oscillating tableau and matchings.

• There should be some kind of definition for descents. Even though it's called 'references', as much as possible you really only want to use them as citations for accreditation, or as an additional reference for finer details/technicalities. Somebody reasonably familiar with the broader subject area should be able to read the docstring and understand what it does without needing to consult outside references.

• In sundaram, I don't know that 'straight oscillating tableau' is standard terminology, and it's never explicitly mentioned that 'straight oscillating tableau' means 'oscillating tableau that starts with the empty shape'. Also, the definition needs to be a little more precise, since 'complement' is ambiguous unless you specify a superset. I would also add an extra line of docstring as an extra paragraph mentioning how if you require the final shape to be empty, you get a bijection to perfect matchings (and include a link to to_perfect_matching).

• to_perfect_matching needs a little more in the docstring (and potentially link to sundaram), and should check that the final shape is empty, and return an error otherwise.

[sagetrac-pfannerer]

Changed branch from u/Bruce/oscillating_tableaux to u/pfannerer/oscillating_tableaux

[sagetrac-pfannerer]

Changed commit from 90673a6 to 376f3ec

[sagetrac-git]

Changed commit from 547c117 to 1e1528a

[sagetrac-git]

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

6f4be3c	Improved documentation and testing thanks to Kevin's comments
453d0ce	More testing
1e1528a	More testing

[sagetrac-git]

Changed commit from 1e1528a to 74b0939

[sagetrac-git]

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

74b0939

Remove import of StandardTableau

[sagetrac-git]

Changed commit from 74b0939 to 8edd839

[sagetrac-git]

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

1511a94	Added more tests
8edd839	More testing

[sagetrac-git]

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

387c0bd	Minor changes in doc strings
0c5bee9	Merge branch 't/25434/pathtableaux' into t/25557/oscillating_tableaux
d714012	Worked on doc strings

[sagetrac-git]

Changed commit from 8edd839 to d714012

[BruceWestbury]

Reviewer: tscrim, chapoton

[sagetrac-git]

Changed commit from d714012 to 396095f

[sagetrac-git]

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

4747bd6	Added catalog.py and changed name
8567a47	Changed _local_rule to local_rule
767405c	Tried to write catalog.py
0942d3b	Edited combinat/all.py all.py etc. to import from catalog
62e7e63	Minor edits
ab20331	Merge branch 't/25434/pathtableaux' into t/25557/oscillating_tableaux
c10f041	Edited all.py catalog.py etc.
396095f	Removed an underscore

[sagetrac-git]

Changed commit from 396095f to 247e3d2

[sagetrac-git]

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

b248c6a	Changed to one line if
29d166f	Merge branch 't/25434/pathtableaux' into t/25557/oscillating_tableaux
247e3d2	Removed .format in Error messages

[mkoeppe]

comment:30

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[mkoeppe]

comment:31

Setting a new milestone for this ticket based on a cursory review.

[mkoeppe]

comment:32

Stalled in needs_review or needs_info; likely won't make it into Sage 9.5.

[fchapoton]

Changed reviewer from tscrim, chapoton to Travis Scrimshaw, Frédéric Chapoton