commit

6_18_2024_catalan/catalan.py

@@ -2,6 +2,7 @@
 
 def parenthesizations(n):
   """
+
   Returns a set of all possible parenthesizations of length n.
 
   Parameters:
@@ -17,9 +18,25 @@ def parenthesizations(n):
   if n == 0:
     return {""}
   else:
-    # TODO
-    pass
+    output = set()
+    parenthesizations_helper("", n, n, output)
+    return output
 
+"""
+res - result string
+open - number of open parenthesis
+close - number of close parenthesis
+output - output set
+"""
+def parenthesizations_helper(res, open, close, output):
+  if open == 0 and close == 0:
+    output.add(res)
+  else:
+    if open > 0:
+      parenthesizations_helper(res + "(", open - 1, close, output)
+    if close != open:
+      parenthesizations_helper(res + ")", open, close - 1, output)             
+
 def product_orders(n):
   """
   Returns a set of all possible ways to multiply of n elements.
@@ -41,8 +58,23 @@ def product_orders(n):
   elif n == 2:
     return {"?*?"}
   else:
-    # TODO
-    pass
+    output = set()
+    return product_orders_helper(n, n)
+
+
+def product_orders_helper(n, overall_length):
+  if n == 1:
+    return {"?"}
+  output = set()
+  for i in range(1, n):
+    for j in product_orders_helper(i, overall_length):
+      for k in product_orders_helper(n - i, overall_length):
+        new_string = j + "*" + k
+        if new_string[0] == "?" or new_string[len(new_string) - 1] == "?":
+          if n < overall_length:
+            new_string = "(" + new_string + ")"
+        output.add(new_string)
+  return output
 
 def permutations_avoiding_231(n):
   """
@@ -58,11 +90,32 @@ def permutations_avoiding_231(n):
   >>> permutations_avoiding_231(4)
   {(1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4), (1, 4, 2, 3), (1, 4, 3, 2), (2, 1, 3, 4), (2, 1, 4, 3), (3, 1, 2, 4), (3, 2, 1, 4), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3), (4, 3, 1, 2), (4, 3, 2, 1)}
   """
+
   if n < 3:
     return set(itertools.permutations(range(1, n+1)))
   else:
-    # TODO
-    pass
+    perms = itertools.permutations(range(1, n+1))
+    output = set()
+    found = False
+    for perm in perms:
+      found = False
+      for i in range(n - 2):
+        first = perm[i]
+        for j in range(i + 1, n - 1):
+          second = perm[j]
+          for k in range(j + 1, n):
+            third = perm[k]
+            if second > first and first > third:
+              found = True
+              break
+          if found:
+            break
+        if found:
+          break
+      if not found:
+        output.add(perm)
+    return output     
+
 
 def triangulations(n):
   """
@@ -84,5 +137,23 @@ def triangulations(n):
   elif n == 3:
     return {tuple()}
   else:
-    pass
-    # TODO
+    vertices = list(range(n - 1))
+    output = set()
+    for i in range (n - 2):
+      output = output.union(triangulation_helper(i, vertices))
+    return output
+
+
+def triangulation_helper(start, vertices):
+  length = len(vertices)
+  vertices = sorted(vertices)
+  output = set()
+  if len(vertices) == 3:
+    return {((vertices[(start) % length], vertices[(start + 1) % length]), (vertices[start % length], vertices[(start + 2) % length]))}
+  else:
+    triangle = ((vertices[(start) % length], vertices[(start + 1) % length]), (vertices[start % length], vertices[(start + 2) % length]))
+    del vertices[(start + 1) % length]
+    output.add(triangle)
+    return output.union(triangulation_helper(start, vertices))
+
+