gh-40245: moving linear_intervals_count to hasse as iterator
    
to make them still faster ; also changed the moved method into an
iterator as this could be convenient

follow-up of previous effort in #40171

Timings :

poset of size 4800 : 18s before ; 13 after

poset of size 5300 : 24.4 s before  and17.4 s after

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: #40245
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/combinat/posets/hasse_diagram.py

@@ -2427,13 +2427,56 @@ def chain_polynomial(self):
         """
         return chain_poly(self._leq_storage)._sage_('q')  # noqa: F821
 
+    def linear_intervals_count(self):
+        """
+        Return the enumeration of linear intervals w.r.t. their cardinality.
+
+        An interval is linear if it is a total order.
+
+        OUTPUT: an iterator of integers
+
+        .. SEEALSO:: :meth:`is_linear_interval`
+
+        EXAMPLES::
+
+            sage: P = posets.BubblePoset(3,3)
+            sage: H = P._hasse_diagram
+            sage: list(H.linear_intervals_count())
+            [245, 735, 438, 144, 24]
+        """
+        if not self:
+            return
+        # precomputation helps for speed:
+        _ = self._leq_storage
+
+        stock = [(x, x, x) for x in self]
+        yield len(stock)
+        exposant = 0
+        while True:
+            exposant += 1
+            next_stock = []
+            short_stock = [(ch[0], ch[2]) for ch in stock]
+            for xmin, cov_xmin, xmax in stock:
+                for y in self.neighbor_out_iterator(xmax):
+                    if exposant == 1:
+                        next_stock.append((xmin, y, y))
+                    elif (cov_xmin, y) in short_stock:
+                        if self.is_linear_interval(xmin, y):
+                            next_stock.append((xmin, cov_xmin, y))
+            if next_stock:
+                yield len(next_stock)
+                stock = next_stock
+            else:
+                break
+
     def is_linear_interval(self, t_min, t_max) -> bool:
         """
         Return whether the interval ``[t_min, t_max]`` is linear.
 
         This means that this interval is a total order.
 
         EXAMPLES::
+
             sage: # needs sage.modules
             sage: P = posets.PentagonPoset()
             sage: H = P._hasse_diagram

src/sage/combinat/posets/posets.py

@@ -2771,31 +2771,7 @@ def linear_intervals_count(self) -> list[int]:
         """
         if not self.cardinality():
             return []
-        # precomputation helps for speed:
-        if self.cardinality() > 60:
-            self.lequal_matrix()
-
-        H = self._hasse_diagram
-        stock = [(x, x, x) for x in H]
-        poly = [len(stock)]
-        exposant = 0
-        while True:
-            exposant += 1
-            next_stock = []
-            short_stock = [(ch[0], ch[2]) for ch in stock]
-            for xmin, cov_xmin, xmax in stock:
-                for y in H.neighbor_out_iterator(xmax):
-                    if exposant == 1:
-                        next_stock.append((xmin, y, y))
-                    elif (cov_xmin, y) in short_stock:
-                        if H.is_linear_interval(xmin, y):
-                            next_stock.append((xmin, cov_xmin, y))
-            if next_stock:
-                poly.append(len(next_stock))
-                stock = next_stock
-            else:
-                break
-        return poly
+        return list(self._hasse_diagram.linear_intervals_count())
 
     def is_linear_interval(self, x, y) -> bool:
         """