gh-37634: The Glaisher-Franklin bijections on integer partitions
    
The Glaisher-Franklin bijections, for a positive integer $s$, map the
set of parts divisible by $s$ to the set of parts which occur at least
$s$ times, see https://www.findstat.org/MapsDatabase/Mp00312 for the
$s=2$ case.

This generalizes and streamlines the code from findstat.
    
URL: #37634
Reported by: Martin Rubey
Reviewer(s): Martin Rubey, Travis Scrimshaw

Author: Release Manager

src/sage/combinat/partition.py

@@ -2485,6 +2485,118 @@ def conjugate(self):
         par = Partitions_n(sum(self))
         return par.element_class(par, conjugate(self))
 
+    def glaisher_franklin(self, s):
+        r"""
+        Apply the Glaisher-Franklin bijection to ``self``.
+
+        The Franklin-Glaisher bijection, with parameter `s`, returns
+        a partition whose set of parts that are repeated at least `s`
+        times equals the set of parts divisible by `s` in ``self``,
+        after dividing each part by `s`.
+
+        INPUT:
+
+        - ``s`` -- positive integer
+
+        EXAMPLES::
+
+            sage: Partition([4, 3, 2, 2, 1]).glaisher_franklin(2)
+            [3, 2, 2, 1, 1, 1, 1, 1]
+
+        TESTS:
+
+        The map preserves the size::
+
+            sage: all(mu.glaisher_franklin(s).size() == n
+            ....:     for n in range(20) for mu in Partitions(n)
+            ....:     for s in range(1, 5))
+            True
+
+        The map is bijective::
+
+            sage: l = [[mu.glaisher_franklin(s)
+            ....:      for n in range(20) for mu in Partitions(n)]
+            ....:     for s in range(1, 5)]
+            sage: all(len(set(ls)) == len(ls) for ls in l)
+            True
+
+        The map transports the statistics::
+
+            sage: d = lambda la, s: set(p / s for p in la if p % s == 0)
+            sage: r = lambda la, s: set(p for p in la if list(la).count(p) >= s)
+            sage: all(d(mu, s) == r(mu.glaisher_franklin(s), s)
+            ....:     for n in range(20) for mu in Partitions(n)
+            ....:     for s in range(1, 5))
+            True
+
+        For `s=2`, the map is known to findstat::
+
+            sage: findmap(Partitions, lambda mu: mu.glaisher_franklin(2))       # optional - internet
+            0: Mp00312 (quality [100])
+        """
+        s = ZZ(s)
+        if s.is_one():
+            return self
+        mu = []
+        for p, m in enumerate(self.to_exp(), 1):
+            if not p % s:
+                mu.extend([p // s] * (m*s))
+            else:
+                for i, v in enumerate(m.digits(s)):
+                    mu.extend([p * s**i]*v)
+
+        P = self.parent()
+        return P.element_class(P, sorted(mu, reverse=True))
+
+    def glaisher_franklin_inverse(self, s):
+        r"""
+        Apply the inverse of the Glaisher-Franklin bijection to ``self``.
+
+        The inverse of the Franklin-Glaisher bijection, with
+        parameter `s`, returns a partition whose set of parts that
+        are divisible by `s`, after dividing each by `s`, equals the
+        equals the set of parts repeated at least `s` times in
+        ``self``.
+
+        INPUT:
+
+        - ``s`` -- positive integer
+
+        EXAMPLES::
+
+            sage: Partition([4, 3, 2, 2, 1]).glaisher_franklin(2)
+            [3, 2, 2, 1, 1, 1, 1, 1]
+            sage: Partition([3, 2, 2, 1, 1, 1, 1, 1]).glaisher_franklin_inverse(2)
+            [4, 3, 2, 2, 1]
+
+        TESTS:
+
+        The map is inverse to :meth:`glaisher_franklin`::
+
+            sage: all(mu.glaisher_franklin(s).glaisher_franklin_inverse(s) == mu
+            ....:     and mu.glaisher_franklin_inverse(s).glaisher_franklin(s) == mu
+            ....:     for n in range(20) for mu in Partitions(n)
+            ....:     for s in range(1, 5))
+            True
+
+        For `s=2`, the map is known to findstat::
+
+            sage: findmap(Partitions, lambda mu: mu.glaisher_franklin_inverse(2))         # optional - internet
+            0: Mp00313 (quality [100])
+        """
+        s = ZZ(s)
+        if s.is_one():
+            return self
+        mu = []
+        for p, m in enumerate(self.to_exp(), 1):
+            p = ZZ(p)
+            mu.extend([p * s] * (m // s))
+            m1, p1 = p.val_unit(s)
+            mu.extend([p1] * ((m % s) * s**m1))
+
+        P = self.parent()
+        return P.element_class(P, sorted(mu, reverse=True))
+
     def suter_diagonal_slide(self, n, exp=1):
         r"""
         Return the image of ``self`` in `Y_n` under Suter's diagonal slide