index 6462a48154a0170fbafec886d38d11c10f10c554..816679d7e0338201995ff0e2aca1e7dc9bb84c85 100644 (file)
@@ -272,6 +272,175 @@ again:
}
}

+/** Generate all bounded combinatorial partitions of an integer n with exactly
+ *  m parts (including zero parts) in non-decreasing order.
+ */
+class partition_generator {
+private:
+       // Partitions n into m parts, not including zero parts.
+       // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
+       // FXT library)
+       struct mpartition2
+       {
+               // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
+               std::vector<int> x;
+               int n;   // n>0
+               int m;   // 0<m<=n
+               mpartition2(unsigned n_, unsigned m_)
+                 : x(m_+1), n(n_), m(m_)
+               {
+                       for (int k=1; k<m; ++k)
+                               x[k] = 1;
+                       x[m] = n - m + 1;
+               }
+               bool next_partition()
+               {
+                       int u = x[m];  // last element
+                       int k = m;
+                       int s = u;
+                       while (--k) {
+                               s += x[k];
+                               if (x[k] + 2 <= u)
+                                       break;
+                       }
+                       if (k==0)
+                               return false;  // current is last
+                       int f = x[k] + 1;
+                       while (k < m) {
+                               x[k] = f;
+                               s -= f;
+                               ++k;
+                       }
+                       x[m] = s;
+                       return true;
+               }
+       } mpgen;
+       int m;  // number of parts 0<m<=n
+       mutable std::vector<int> partition;  // current partition
+public:
+       partition_generator(unsigned n_, unsigned m_)
+         : mpgen(n_, 1), m(m_), partition(m_)
+       { }
+       // returns current partition in non-decreasing order, padded with zeros
+       const std::vector<int>& current() const
+       {
+               for (int i = 0; i < m - mpgen.m; ++i)
+                       partition[i] = 0;  // pad with zeros
+
+               for (int i = m - mpgen.m; i < m; ++i)
+                       partition[i] = mpgen.x[i - m + mpgen.m + 1];
+
+               return partition;
+       }
+       bool next()
+       {
+               if (!mpgen.next_partition()) {
+                       if (mpgen.m == m || mpgen.m == mpgen.n)
+                               return false;  // current is last
+                       // increment number of parts
+                       mpgen = mpartition2(mpgen.n, mpgen.m + 1);
+               }
+               return true;
+       }
+};
+
+/** Generate all compositions of a partition of an integer n, starting with the
+ *  compositions which has non-decreasing order.
+ */
+class composition_generator {
+private:
+       // Generates all distinct permutations of a multiset.
+       // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
+       // Multiset Permutations using a Constant Number of Variables by Prefix
+       // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
+       struct coolmulti {
+               // element of singly linked list
+               struct element {
+                       int value;
+                       element* next;
+                       element(int val, element* n)
+                         : value(val), next(n) {}
+                       ~element()
+                       {   // recurses down to the end of the singly linked list
+                               delete next;
+                       }
+               };
+               element *head, *i, *after_i;
+               // NB: Partition must be sorted in non-decreasing order.
+               explicit coolmulti(const std::vector<int>& partition)
+                 : head(nullptr), i(nullptr), after_i(nullptr)
+               {
+                       for (unsigned n = 0; n < partition.size(); ++n) {
+                               head = new element(partition[n], head);
+                               if (n <= 1)
+                                       i = head;
+                       }
+                       after_i = i->next;
+               }
+               ~coolmulti()
+               {   // deletes singly linked list
+                       delete head;
+               }
+               void next_permutation()
+               {
+                       element *before_k;
+                       if (after_i->next != nullptr && i->value >= after_i->next->value)
+                               before_k = after_i;
+                       else
+                               before_k = i;
+                       element *k = before_k->next;
+                       before_k->next = k->next;
+                       k->next = head;
+                       if (k->value < head->value)
+                               i = k;
+                       after_i = i->next;
+                       head = k;
+               }
+               bool finished() const
+               {
+                       return after_i->next == nullptr && after_i->value >= head->value;
+               }
+       } cmgen;
+       bool atend;  // needed for simplifying iteration over permutations
+       bool trivial;  // likewise, true if all elements are equal
+       mutable std::vector<int> composition;  // current compositions
+public:
+       explicit composition_generator(const std::vector<int>& partition)
+         : cmgen(partition), atend(false), trivial(true), composition(partition.size())
+       {
+               for (unsigned i=1; i<partition.size(); ++i)
+                       trivial = trivial && (partition[0] == partition[i]);
+       }
+       const std::vector<int>& current() const
+       {
+               coolmulti::element* it = cmgen.head;
+               size_t i = 0;
+               while (it != nullptr) {
+                       composition[i] = it->value;
+                       it = it->next;
+                       ++i;
+               }
+               return composition;
+       }
+       bool next()
+       {
+               // This ugly contortion is needed because the original coolmulti
+               // algorithm requires code duplication of the payload procedure,
+               // one before the loop and one inside it.
+               if (trivial || atend)
+                       return false;
+               cmgen.next_permutation();
+               atend = cmgen.finished();
+               return true;
+       }
+};
+
+/** Compute the multinomial coefficient n!/(p1!*p2!*...*pk!) where
+ *  n = p1+p2+...+pk, i.e. p is a partition of n.
+ */
+const numeric
+multinomial_coefficient(const std::vector<int> & p);
+

// Collection of `construct on first use' wrappers for safely avoiding
// internal object replication without running into the `static