Trac #14126: Count Number of Linear Extensions of a Poset

Posets appear to have no mechanism to count the number of linear
extensions other than computing all the linear extensions and finding
the length of that list. Using recursion, one can count the number of
linear extensions without having to generate all the linear extensions.

URL: https://trac.sagemath.org/14126
Reported by: csar
Ticket author(s): Jori Mäntysalo
Reviewer(s): Frédéric Chapoton

src/sage/combinat/posets/linear_extensions.py

@@ -432,6 +432,35 @@ def poset(self):
         """
         return self._poset
 
+    def cardinality(self):
+        """
+        Return the number of linear extensions.
+
+        EXAMPLES::
+
+            sage: N = Poset({0: [2, 3], 1: [3]})
+            sage: N.linear_extensions().cardinality()
+            5
+
+        TESTS::
+
+            sage: Poset().linear_extensions().cardinality()
+            1
+            sage: Posets.ChainPoset(1).linear_extensions().cardinality()
+            1
+        """
+        from sage.rings.integer import Integer
+
+        H = self._poset.order_ideals_lattice(as_ideals=False)._hasse_diagram
+        L = H.level_sets()
+        c = [0] * H.order()
+        for l in L[0]:
+            c[l] = 1
+        for lev in L[1:]:
+            for l in lev:
+                c[l] = sum(c[i] for i in H.lower_covers_iterator(l))
+        return Integer(sum(c[i] for i in H.sinks()))
+
     def __iter__(self):
         r"""
         Iterates through the linear extensions of the underlying poset.