gh-38795: switch to nauty for generating posets

as there is now a direct method in `nauty` and we have added the
interface previously

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

URL: #38795
Reported by: Frédéric Chapoton
Reviewer(s): David Coudert, Frédéric Chapoton, Martin Rubey

Author: Release Manager

build/pkgs/configure/checksums.ini

@@ -1,3 +1,3 @@
 tarball=configure-VERSION.tar.gz
-sha1=140d921780212198287a0d626b8c9655d4408f7a
-sha256=4f82ec8bdb67c4dffd43daddd5160e8f9c624f6a5cf16c4128bfab2e5ddca459
+sha1=2b8c8181575563ae16afca012c2188b31f3e8bb7
+sha256=77d20639b03b1c125d6ccf88ebb2cb0202b32059dff4bf38ca05db229913f639

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-2b4fe39e420df022d34a67fbc820252618e15f83
+a1d9c1e13576bf777e95ba50d51d81ee09d2a0c9

src/sage/combinat/posets/posets.py

@@ -8884,11 +8884,11 @@ class FinitePosets_n(UniqueRepresentation, Parent):
         sage: P.cardinality()
         5
         sage: for p in P: print(p.cover_relations())
-        []
-        [[1, 2]]
+        [[1, 2], [0, 2]]
         [[0, 1], [0, 2]]
+        [[0, 2]]
         [[0, 1], [1, 2]]
-        [[1, 2], [0, 2]]
+        []
     """
 
     def __init__(self, n) -> None:
@@ -8930,27 +8930,40 @@ def __contains__(self, P) -> bool:
         return P in FinitePosets() and P.cardinality() == self._n
 
     def __iter__(self):
-        """
+        r"""
         Return an iterator of representatives of the isomorphism classes
         of finite posets of a given size.
 
         .. NOTE::
 
-            This uses the DiGraph iterator as a backend to construct
-            transitively-reduced, acyclic digraphs.
+            If the size `n\leq 16`, this uses an iterator from
+            ``nauty`` as a backend to construct only transitively-reduced,
+            acyclic digraphs. Otherwise it uses a slow naive iterator,
+            as the ``nauty`` iterator is not available.
 
         EXAMPLES::
 
             sage: P = Posets(2)
             sage: list(P)
             [Finite poset containing 2 elements, Finite poset containing 2 elements]
+
+        TESTS::
+
+            sage: it = iter(Posets(17))
+            sage: next(it)
+            Finite poset containing 17 elements
         """
-        from sage.graphs.digraph_generators import DiGraphGenerators
-        for dig in DiGraphGenerators()(self._n, is_poset):
+        if self._n <= 16:
+            it = digraphs.nauty_posetg(f"{self._n} o")
+        else:
+            it = digraphs(self._n, is_poset)
+
+        for dig in it:
             # We need to relabel the digraph since range(self._n) must be a linear
             # extension. Too bad we need to compute this again. TODO: Fix this.
-            label_dict = dict(zip(dig.topological_sort(), range(dig.order())))
-            yield FinitePoset(dig.relabel(label_dict, inplace=False))
+            label_dict = dict(zip(dig.topological_sort(), range(self._n)))
+            dig.relabel(label_dict, inplace=True)
+            yield FinitePoset(dig)
 
     def cardinality(self, from_iterator=False):
         r"""

src/sage/combinat/tutorial.py

@@ -844,15 +844,15 @@
 
 Partial orders on a set of `8` elements, up to isomorphism::
 
-    sage: C = Posets(8); C
-    Posets containing 8 elements
+    sage: C = Posets(7); C
+    Posets containing 7 elements
     sage: C.cardinality()
-    16999
+    2045
 
 ::
 
     sage: C.unrank(20).plot()
-    Graphics object consisting of 20 graphics primitives
+    Graphics object consisting of ... graphics primitives
 
 .. image:: ../../media/a_poset.png
 

src/sage/databases/findstat.py

@@ -4002,16 +4002,17 @@ def _finite_lattices(n):
     TESTS::
 
         sage: from sage.databases.findstat import _finite_lattices
-        sage: [L.cover_relations() for L in _finite_lattices(4)]
-        [[['bottom', 0], ['bottom', 1], [0, 'top'], [1, 'top']],
-         [['bottom', 0], [0, 1], [1, 'top']]]
+        sage: sorted((L.cover_relations() for L in _finite_lattices(4)),
+        ....:        key=len)
+        [[['bottom', 0], [0, 1], [1, 'top']],
+         [['bottom', 0], ['bottom', 1], [0, 'top'], [1, 'top']]]
     """
     if n <= 2:
         for P in Posets(n):
             if P.is_lattice():
                 yield LatticePoset(P)
     else:
-        for P in Posets(n-2):
+        for P in Posets(n - 2):
             Q = P.with_bounds()
             if Q.is_lattice():
                 yield LatticePoset(Q)