Salvage work on gammoids

[gmou3]

Add matroid classes Gammoid and TransversalMatroid. Transversal matroids are also gammoids.

From the archived sagetrac-mirror repository (branches public/matroids/transversal_matroids-23536 and u/zgershkoff/gammoids).

Resolves #23536, resolves #23601, resolves #23628.

[github-actions]

Documentation preview for this PR (built with commit 61f6748; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[kwankyu]

Thanks. Math aside, the code looks good to me.

I believe the math was also double-checked by you and the original author.

[tobiasdiez]

As @collares pointed out in #38749, some of the tests fail with Python 3.12: https://github.com/sagemath/sage/actions/runs/12979684201/job/36195724175

File "src/sage/matroids/transversal_matroid.pyx", line 701, in sage.matroids.transversal_matroid.TransversalMatroid.transversal_extension
Failed example:
    M2 = M.transversal_extension(element='d', newset=True)
Exception raised:
    Traceback (most recent call last):
      File "/Users/runner/work/sage/sage/src/sage/doctest/forker.py", line 728, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/Users/runner/work/sage/sage/src/sage/doctest/forker.py", line 1152, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.matroids.transversal_matroid.TransversalMatroid.transversal_extension[3]>", line 1, in <module>
        M2 = M.transversal_extension(element='d', newset=True)
             ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
      File "sage/matroids/transversal_matroid.pyx", line 671, in sage.matroids.transversal_matroid.TransversalMatroid.transversal_extension
        cpdef transversal_extension(self, element=None, newset=False, sets=[]):
      File "sage/matroids/transversal_matroid.pyx", line 792, in sage.matroids.transversal_matroid.TransversalMatroid.transversal_extension
        return TransversalMatroid(new_sets, groundset, labels)
      File "sage/matroids/transversal_matroid.pyx", line 181, in sage.matroids.transversal_matroid.TransversalMatroid.__init__
        raise ValueError("set labels cannot be element labels")
    ValueError: set labels cannot be element labels

Can you please have a look and fix these? Thanks!

[collares]

Quoting @dcoudert in #39266:

May be the error is due to the declaration cpdef transversal_extension(self, element=None, newset=False, sets=[]): in src/sage/matroids/transversal_matroid.pyx. Using None seems more appropriate.

[gmou3]

Thanks for the reporting. I figured out that there was a bug when choosing a new label for the newly created set.
See #39405.