gh-36830: improved integer vectors efficiency -Enhancement
    
Fixes #36816

1)  **Added** the **cardinality function** using stars and bars to the
**IntegerVector_nk** class
2) Fixed the errors in IntegerVectors_n and IntegerVectors_k now
**IntegerVectors_n(0)cardinality() and IntegerVectors_k(0).cardinality()
returns 1 and both of them are Finite EnumeratedSets**
    
URL: #36830
Reported by: Aman Moon
Reviewer(s): Aman Moon, Jukka Kohonen, Martin Rubey

Author: Release Manager

src/sage/combinat/integer_vector.py

@@ -48,7 +48,7 @@
 from sage.rings.integer import Integer
 
 
-def is_gale_ryser(r,s):
+def is_gale_ryser(r, s):
     r"""
     Tests whether the given sequences satisfy the condition
     of the Gale-Ryser theorem.
@@ -314,20 +314,20 @@ def gale_ryser_theorem(p1, p2, algorithm="gale",
     """
     from sage.matrix.constructor import matrix
 
-    if not is_gale_ryser(p1,p2):
+    if not is_gale_ryser(p1, p2):
         return False
 
-    if algorithm == "ryser": # ryser's algorithm
+    if algorithm == "ryser":  # ryser's algorithm
         from sage.combinat.permutation import Permutation
 
         # Sorts the sequences if they are not, and remembers the permutation
         # applied
-        tmp = sorted(enumerate(p1), reverse=True, key=lambda x:x[1])
+        tmp = sorted(enumerate(p1), reverse=True, key=lambda x: x[1])
         r = [x[1] for x in tmp]
         r_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()]
         m = len(r)
 
-        tmp = sorted(enumerate(p2), reverse=True, key=lambda x:x[1])
+        tmp = sorted(enumerate(p2), reverse=True, key=lambda x: x[1])
         s = [x[1] for x in tmp]
         s_permutation = [x-1 for x in Permutation([x[0]+1 for x in tmp]).inverse()]
 
@@ -340,12 +340,12 @@ def gale_ryser_theorem(p1, p2, algorithm="gale",
                 k = i + 1
                 while k < m and r[i] == r[k]:
                     k += 1
-                if t >= k - i: # == number rows of the same length
+                if t >= k - i:  # == number rows of the same length
                     for j in range(i, k):
                         r[j] -= 1
                         c[j] = 1
                     t -= k - i
-                else: # Remove the t last rows of that length
+                else:  # Remove the t last rows of that length
                     for j in range(k-t, k):
                         r[j] -= 1
                         c[j] = 1
@@ -366,17 +366,17 @@ def gale_ryser_theorem(p1, p2, algorithm="gale",
         k1, k2 = len(p1), len(p2)
         p = MixedIntegerLinearProgram(solver=solver)
         b = p.new_variable(binary=True)
-        for (i,c) in enumerate(p1):
-            p.add_constraint(p.sum([b[i,j] for j in range(k2)]) == c)
-        for (i,c) in enumerate(p2):
-            p.add_constraint(p.sum([b[j,i] for j in range(k1)]) == c)
+        for (i, c) in enumerate(p1):
+            p.add_constraint(p.sum([b[i, j] for j in range(k2)]) == c)
+        for (i, c) in enumerate(p2):
+            p.add_constraint(p.sum([b[j, i] for j in range(k1)]) == c)
         p.set_objective(None)
         p.solve()
         b = p.get_values(b, convert=ZZ, tolerance=integrality_tolerance)
         M = [[0]*k2 for i in range(k1)]
         for i in range(k1):
             for j in range(k2):
-                M[i][j] = b[i,j]
+                M[i][j] = b[i, j]
         return matrix(M)
 
     else:
@@ -780,6 +780,43 @@ def __contains__(self, x):
                 return False
         return True
 
+    def _unrank_helper(self, x, rtn):
+        """
+        Return the element at rank ``x`` by iterating through all integer vectors beginning with ``rtn``.
+
+        INPUT:
+
+        - ``x`` - a nonnegative integer
+        - ``rtn`` - a list of nonnegative integers
+
+
+        EXAMPLES::
+
+            sage: IV = IntegerVectors(k=5)
+            sage: IV._unrank_helper(10, [2,0,0,0,0])
+            [1, 0, 0, 0, 1]
+
+            sage: IV = IntegerVectors(n=7)
+            sage: IV._unrank_helper(100, [7,0,0,0])
+            [2, 0, 0, 5]
+
+            sage: IV = IntegerVectors(n=12, k=7)
+            sage: IV._unrank_helper(1000, [12,0,0,0,0,0,0])
+            [5, 3, 1, 1, 1, 1, 0]
+        """
+        ptr = 0
+        while True:
+            current_rank = self.rank(rtn)
+            if current_rank < x:
+                rtn[ptr+1] = rtn[ptr]
+                rtn[ptr] = 0
+                ptr += 1
+            elif current_rank > x:
+                rtn[ptr] -= 1
+                rtn[ptr-1] += 1
+            else:
+                return self._element_constructor_(rtn)
+
 
 class IntegerVectors_all(UniqueRepresentation, IntegerVectors):
     """
@@ -839,7 +876,10 @@ def __init__(self, n):
             sage: TestSuite(IV).run()
         """
         self.n = n
-        IntegerVectors.__init__(self, category=InfiniteEnumeratedSets())
+        if self.n == 0:
+            IntegerVectors.__init__(self, category=EnumeratedSets())
+        else:
+            IntegerVectors.__init__(self, category=InfiniteEnumeratedSets())
 
     def _repr_(self):
         """
@@ -898,6 +938,68 @@ def __contains__(self, x):
             return False
         return sum(x) == self.n
 
+    def rank(self, x):
+        """
+        Return the rank of a given element.
+
+        INPUT:
+
+        - ``x`` -- a list with ``sum(x) == n``
+
+        EXAMPLES::
+
+            sage: IntegerVectors(n=5).rank([5,0])
+            1
+            sage: IntegerVectors(n=5).rank([3,2])
+            3
+        """
+        if sum(x) != self.n:
+            raise ValueError("argument is not a member of IntegerVectors({},{})".format(self.n, None))
+
+        n, k, s = self.n, len(x), 0
+        r = binomial(k + n - 1, n + 1)
+        for i in range(k - 1):
+            s += x[k - 1 - i]
+            r += binomial(s + i, i + 1)
+        return r
+
+    def unrank(self, x):
+        """
+        Return the element at given rank x.
+
+        INPUT:
+
+        - ``x`` -- an integer.
+
+        EXAMPLES::
+
+            sage: IntegerVectors(n=5).unrank(2)
+            [4, 1]
+            sage: IntegerVectors(n=10).unrank(10)
+            [1, 9]
+        """
+        rtn = [self.n]
+        while self.rank(rtn) <= x:
+            rtn.append(0)
+        rtn.pop()
+
+        return IntegerVectors._unrank_helper(self, x, rtn)
+
+    def cardinality(self):
+        """
+        Return the cardinality of ``self``.
+
+        EXAMPLES::
+
+            sage: IntegerVectors(n=0).cardinality()
+            1
+            sage: IntegerVectors(n=10).cardinality()
+            +Infinity
+        """
+        if self.n == 0:
+            return Integer(1)
+        return PlusInfinity()
+
 
 class IntegerVectors_k(UniqueRepresentation, IntegerVectors):
     """
@@ -912,7 +1014,10 @@ def __init__(self, k):
             sage: TestSuite(IV).run()
         """
         self.k = k
-        IntegerVectors.__init__(self, category=InfiniteEnumeratedSets())
+        if self.k == 0:
+            IntegerVectors.__init__(self, category=EnumeratedSets())
+        else:
+            IntegerVectors.__init__(self, category=InfiniteEnumeratedSets())
 
     def _repr_(self):
         """
@@ -968,6 +1073,75 @@ def __contains__(self, x):
             return False
         return len(x) == self.k
 
+    def rank(self, x):
+        """
+        Return the rank of a given element.
+
+        INPUT:
+
+        - ``x`` -- a list with ``len(x) == k``
+
+        EXAMPLES::
+
+            sage: IntegerVectors(k=5).rank([0,0,0,0,0])
+            0
+            sage: IntegerVectors(k=5).rank([1,1,0,0,0])
+            7
+        """
+        if len(x) != self.k:
+            raise ValueError("argument is not a member of IntegerVectors({},{})".format(None, self.k))
+
+        n, k, s = sum(x), self.k, 0
+        r = binomial(n + k - 1, k)
+        for i in range(k - 1):
+            s += x[k - 1 - i]
+            r += binomial(s + i, i + 1)
+        return r
+
+    def unrank(self, x):
+        """
+        Return the element at given rank x.
+
+        INPUT:
+
+        - ``x`` -- an integer such that x < self.cardinality()``
+
+        EXAMPLES::
+
+            sage: IntegerVectors(k=5).unrank(10)
+            [1, 0, 0, 0, 1]
+            sage: IntegerVectors(k=5).unrank(15)
+            [0, 0, 2, 0, 0]
+            sage: IntegerVectors(k=0).unrank(0)
+            []
+        """
+        if self.k == 0 and x != 0:
+            raise IndexError(f"Index {x} is out of range for the IntegerVector.")
+        rtn = [0]*self.k
+        if self.k == 0 and x == 0:
+            return rtn
+
+        while self.rank(rtn) <= x:
+            rtn[0] += 1
+        rtn[0] -= 1
+
+        return IntegerVectors._unrank_helper(self, x, rtn)
+
+    def cardinality(self):
+        """
+        Return the cardinality of ``self``.
+
+        EXAMPLES::
+
+            sage: IntegerVectors(k=0).cardinality()
+            1
+            sage: IntegerVectors(k=10).cardinality()
+            +Infinity
+        """
+        if self.k == 0:
+            return Integer(1)
+        return PlusInfinity()
+
 
 class IntegerVectors_nk(UniqueRepresentation, IntegerVectors):
     """
@@ -1010,11 +1184,11 @@ def _list_rec(self, n, k):
         res = []
 
         if k == 1:
-            return [ (n, ) ]
+            return [(n, )]
 
         for nbar in range(n + 1):
             n_diff = n - nbar
-            for rest in self._list_rec( nbar , k - 1):
+            for rest in self._list_rec(nbar, k - 1):
                 res.append((n_diff,) + rest)
         return res
 
@@ -1153,17 +1327,49 @@ def rank(self, x):
         if x not in self:
             raise ValueError("argument is not a member of IntegerVectors({},{})".format(self.n, self.k))
 
-        n = self.n
-        k = self.k
-
-        r = 0
+        k, s, r = self.k, 0, 0
         for i in range(k - 1):
-            k -= 1
-            n -= x[i]
-            r += binomial(k + n - 1, k)
-
+            s += x[k - 1 - i]
+            r += binomial(s + i, i + 1)
         return r
 
+    def unrank(self, x):
+        """
+        Return the element at given rank x.
+
+        INPUT:
+
+        - ``x`` -- an integer such that ``x < self.cardinality()``
+
+        EXAMPLES::
+
+            sage: IntegerVectors(4,5).unrank(30)
+            [1, 0, 1, 0, 2]
+            sage: IntegerVectors(2,3).unrank(5)
+            [0, 0, 2]
+        """
+        if x >= self.cardinality():
+            raise IndexError(f"Index {x} is out of range for the IntegerVector.")
+        rtn = [0]*self.k
+        rtn[0] = self.n
+        return IntegerVectors._unrank_helper(self, x, rtn)
+
+    def cardinality(self):
+        """
+        Return the cardinality of ``self``.
+
+        EXAMPLES::
+
+            sage: IntegerVectors(3,5).cardinality()
+            35
+            sage: IntegerVectors(99, 3).cardinality()
+            5050
+            sage: IntegerVectors(10^9 - 1, 3).cardinality()
+            500000000500000000
+        """
+        n, k = self.n, self.k
+        return Integer(binomial(n + k - 1, n))
+
 
 class IntegerVectors_nnondescents(UniqueRepresentation, IntegerVectors):
     r"""
@@ -1320,11 +1526,11 @@ def __init__(self, n=None, k=None, **constraints):
                 category = FiniteEnumeratedSets()
             else:
                 category = EnumeratedSets()
-        elif k is not None and 'max_part' in constraints: # n is None
+        elif k is not None and 'max_part' in constraints:  # n is None
             category = FiniteEnumeratedSets()
         else:
             category = EnumeratedSets()
-        IntegerVectors.__init__(self, category=category) # placeholder category
+        IntegerVectors.__init__(self, category=category)  # placeholder category
 
     def _repr_(self):
         """