This is the set of all subsets of a multiset, which are themselves multisets, i.e., the power set of a multiset.

For example, the power set of the multiset [a, b, b, c] consists of the sets:

()
(c)
(b)
(b, c)
(b, b)
(b, b, c)
(a)
(a, c)
(a, b)
(a, b, c)
(a, b, b)
(a, b, b, c)

The set of all subsets of a particular size are combinations of a multiset.

The multiset and its subsets are represented as a vector containing, for each element, the count of its occurrences in the multiset. For example, the multiset [a, b, b, c] is represented as [1, 2, 1]. This is similar to the characteristic vector used for power set, but with counts rather than boolean values.

The correspondences for the subsets are then as follows:

[0, 0, 0] = ()
[0, 0, 1] = (c)
[0, 1, 0] = (b)
[0, 1, 1] = (b, c)
[0, 2, 0] = (b, b)
[0, 2, 1] = (b, b, c)
[1, 0, 0] = (a)
[1, 0, 1] = (a, c)
[1, 1, 0] = (a, b)
[1, 1, 1] = (a, b, c)
[1, 2, 0] = (a, b, b)
[1, 2, 1] = (a, b, b, c)

The algorithm can then simply count from [0, 0, 0] to [1, 2, 1], using the values in the source multiset as the upper limit for the value of an element.

Note that this algorithm does not produce the subsets in lexicographic order.

unsigned int next_multiset_subset(const unsigned int *multiset, unsigned int *ar, size_t n)
{
unsigned int changed = 0;
int i;
for (i = n - 1; i >= 0 && !changed; i--) {
if (ar[i] < multiset[i]) {
/* Increment */
ar[i]++;
changed = 1;
}
else {
/* Roll over */
ar[i] = 0;
}
}
if (!changed) {
/* Reset to first combination */
for (i = 0; i < n; i++) {
ar[i] = 0;
}
}
return changed;
}

Here is an example program:

#include <stdio.h>
#include <multiset-subset.h>
static void print_list(const unsigned int *ar, size_t len, FILE *fptr)
{
unsigned int i;
fputc('(', fptr);
for (i = 0; i < len; i++) {
fprintf(fptr, "%d", ar[i]);
if (i < len - 1) {
fputs(", ", fptr);
}
}
fputc(')', fptr);
}
int main(void)
{
unsigned int multiset[] = {1, 2, 1};
const size_t n = sizeof(multiset) / sizeof(unsigned int);
unsigned int numbers[] = {0, 0, 0};
do {
print_list(numbers, n, stdout);
putchar('\n');
} while (next_multiset_subset(multiset, numbers, n));
return 0;
}

Here is a second example that prints the elements of the multisets:

#include <stdio.h>
#include <multiset-subset.h>
static void print_array(const unsigned int *ar, size_t len, FILE *fptr)
{
unsigned int i;
fputc('[', fptr);
for (i = 0; i < len; i++) {
fprintf(fptr, "%d", ar[i]);
if (i < len - 1) {
fputs(", ", fptr);
}
}
fputc(']', fptr);
}
static void print_multiset_subset(const unsigned int *ar, size_t len,
const void **elements, printfn print, FILE *fptr)
{
unsigned int i, started = 0;
fputc('(', fptr);
for (i = 0; i < len; i++) {
unsigned int j;
for (j = 0; j < ar[i]; j++) {
if (started) {
fputs(", ", fptr);
}
print(elements[i], fptr);
started = 1;
}
}
fputc(')', fptr);
}
int main(void)
{
unsigned int multiset[] = {1, 2, 1};
const size_t n = sizeof(multiset) / sizeof(unsigned int);
char *elements[] = {"a", "b", "c"};
unsigned int numbers[] = {0, 0, 0};
do {
print_array(numbers, n, stdout);
printf(" = ");
print_multiset_subset(numbers, n, (void*)elements, (printfn)fputs, stdout);
putchar('\n');
} while (next_multiset_subset(multiset, numbers, n));
return 0;
}