renaming combinatorial species

[mantepse]

The more thorough renaming of species #39521 introduced a major performance issue: whenever we create a species of not very small degree, eg. AtomicSpecies("X")(SymmetricGroup(26)), all set-like species of this degree are created. The number of these is https://oeis.org/A007650. Unless n is prime, this number may be very large. For example, there are more than one million set like species of degree 66.

[mantepse]

One idea is to not rename at creation, but only when computing _repr_. That is

diff --git a/src/sage/rings/species.py b/src/sage/rings/species.py
index 4d10d8c5cb8..bd675d666a0 100644
--- a/src/sage/rings/species.py
+++ b/src/sage/rings/species.py
@@ -285,6 +285,11 @@ class AtomicSpeciesElement(WithEqualityById,
         self._mc = tuple([len(v) for v in self._dompart])
         self._tc = sum(self._mc)
 
+    def __repr__(self):
+        if self._tc not in self.parent()._renamed:
+            self.parent()._rename(self._tc)
+        return super().__repr__()
+
     def _repr_(self):
         r"""
         Return a string representation of ``self``.
@@ -613,8 +618,6 @@ class AtomicSpecies(UniqueRepresentation, Parent):
                 raise ValueError(f"values of pi (={pi.values()}) must partition the domain of G (={G.domain()})")
         dompart = [pi.get(s, []) for s in range(self._arity)]
         elm = self.element_class(self, G, dompart)
-        if elm._tc not in self._renamed:
-            self._rename(elm._tc)
         return elm
 
     def _rename(self, n):

[mantepse]

Another, complementary idea is to rename only optionally, perhaps up to some fixed degree.