Trac #19215: Posets: Add is_series_parallel()

Add a function to see if a poset is series-parallel (or "N-free").

First wait for #19659 to get closed. Series-parallel decomposition is
just recursive applying of `connected_components()` and
`ordinal_sum_decomposition()`.

URL: http://trac.sagemath.org/19215
Reported by: jmantysalo
Ticket author(s): Jori Mäntysalo
Reviewer(s): Frédéric Chapoton

src/sage/combinat/posets/posets.py

@@ -71,6 +71,7 @@
     :meth:`~FinitePoset.is_graded` | Return ``True`` if all maximal chains of the poset has same length.
     :meth:`~FinitePoset.is_ranked` | Return ``True`` if the poset has a rank function.
     :meth:`~FinitePoset.is_rank_symmetric` | Return ``True`` if the poset is rank symmetric.
+    :meth:`~FinitePoset.is_series_parallel` | Return ``True`` if the poset can be built by ordinal sums and disjoint unions.
     :meth:`~FinitePoset.is_eulerian` | Return ``True`` if the poset is Eulerian.
     :meth:`~FinitePoset.is_incomparable_chain_free` | Return ``True`` if the poset is (m+n)-free.
     :meth:`~FinitePoset.is_slender` | Return ``True`` if the poset is slender.
@@ -2549,9 +2550,40 @@ def is_connected(self):
         """
         return self._hasse_diagram.is_connected()
 
+    def is_series_parallel(self):
+        """
+        Return ``True`` if the poset is series-parallel, and ``False``
+        otherwise.
+
+        A poset is *series-parallel* if it can be built up from one-element
+        posets using the operations of disjoint union and ordinal
+        sum. This is also called *N-free* property: every poset that is not
+        series-parallel contains a subposet isomorphic to the 4-element
+        N-shaped poset where `a > c, d` and `b > d`.
+
+        See :wikipedia:`Series-parallel partial order`.
+
+        EXAMPLES::
+
+            sage: VA = Poset({1: [2, 3], 4: [5], 6: [5]})
+            sage: VA.is_series_parallel()
+            True
+            sage: big_N = Poset({1: [2, 4], 2: [3], 4:[7], 5:[6], 6:[7]})
+            sage: big_N.is_series_parallel()
+            False
+
+        TESTS::
+
+            sage: Poset().is_series_parallel()
+            True
+        """
+        # TODO: Add series-parallel decomposition later.
+        N = Poset({0: [2, 3], 1: [3]})
+        return not self.has_isomorphic_subposet(N)
+
     def is_EL_labelling(self, f, return_raising_chains=False):
         r"""
-        Returns ``True`` if ``f`` is an EL labelling of ``self``.
+        Return ``True`` if ``f`` is an EL labelling of ``self``.
 
         A labelling `f` of the edges of the Hasse diagram of a poset
         is called an EL labelling (edge lexicographic labelling) if