Trac #34368: implement the F,H,M triangles

as useful invariants of posets and simplicial complexes

URL: https://trac.sagemath.org/34368
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw, John Palmieri

src/doc/en/reference/combinat/module_list.rst

@@ -360,6 +360,7 @@ Comprehensive Module List
     sage/combinat/tamari_lattices
     sage/combinat/tiling
     sage/combinat/tools
+    sage/combinat/triangles_FHM
     sage/combinat/tuple
     sage/combinat/tutorial
     sage/combinat/vector_partition

src/sage/combinat/posets/posets.py

@@ -184,6 +184,7 @@
     :meth:`~FinitePoset.flag_h_polynomial` | Return the flag h-polynomial of the poset.
     :meth:`~FinitePoset.order_polynomial` | Return the order polynomial of the poset.
     :meth:`~FinitePoset.zeta_polynomial` | Return the zeta polynomial of the poset.
+    :meth:`~FinitePoset.M_triangle` | Return the M-triangle of the poset.
     :meth:`~FinitePoset.kazhdan_lusztig_polynomial` | Return the Kazhdan-Lusztig polynomial of the poset.
     :meth:`~FinitePoset.coxeter_polynomial` | Return the characteristic polynomial of the Coxeter transformation.
     :meth:`~FinitePoset.degree_polynomial` | Return the generating polynomial of degrees of vertices in the Hasse diagram.
@@ -7259,6 +7260,47 @@ def zeta_polynomial(self):
             f = g[n] + f / n
         return f
 
+    def M_triangle(self):
+        r"""
+        Return the M-triangle of the poset.
+
+        The poset is expected to be graded.
+
+        OUTPUT:
+
+        an :class:`~sage.combinat.triangles_FHM.M_triangle`
+
+        The M-triangle is the generating polynomial of the Möbius numbers
+
+        .. MATH::
+
+            M(x, y)=\sum_{a \leq b} \mu(a,b) x^{|a|}y^{|b|} .
+
+        EXAMPLES::
+
+            sage: P = posets.DiamondPoset(5)
+            sage: P.M_triangle()
+            M: x^2*y^2 - 3*x*y^2 + 3*x*y + 2*y^2 - 3*y + 1
+
+        TESTS::
+
+            sage: P = posets.PentagonPoset()
+            sage: P.M_triangle()
+            Traceback (most recent call last):
+            ...
+            ValueError: the poset is not graded
+        """
+        from sage.combinat.triangles_FHM import M_triangle
+        hasse = self._hasse_diagram
+        rk = hasse.rank_function()
+        if rk is None:
+            raise ValueError('the poset is not graded')
+        x, y = polygen(ZZ, 'x,y')
+        p = sum(hasse.moebius_function(a, b) * x**rk(a) * y**rk(b)
+                for a in hasse
+                for b in hasse.principal_order_filter(a))
+        return M_triangle(p)
+
     def f_polynomial(self):
         r"""
         Return the `f`-polynomial of the poset.
@@ -7296,12 +7338,17 @@ def f_polynomial(self):
             sage: P = Poset({2: []})
             sage: P.f_polynomial()
             1
+
+            sage: P = Poset({2:[1,3]})
+            sage: P.f_polynomial()
+            Traceback (most recent call last):
+            ...
+            ValueError: the poset is not bounded
         """
         q = polygen(ZZ, 'q')
-        one = q.parent().one()
         hasse = self._hasse_diagram
         if len(hasse) == 1:
-            return one
+            return q.parent().one()
         maxi = hasse.top()
         mini = hasse.bottom()
         if mini is None or maxi is None: