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A multiset is an unordered collection of items where – unlike with a set – an item can repeat. For example (we will indicate the items by letters),

a a b b b c c d d d

The order of the items doesn’t matter.Problem: Given a multiset find all its multisubsets, that is, all multisets that can be made from its items:

a c d

b b b c

a c c d d

etc.

b b b c

a c c d d

etc.

The key to the solution is a certain way of representing multisets. First group the same items together, which can be done by putting them in alphabetical order as above. After doing that the multiset can be represented by the successive counts of items. In the example above the ten items come in four blocks,

2 3 2 3

two A’s, three B’s, two C’s, and three D’s.Then we can use four numbers r

The total number of multisets will be (n

This generalizes the following problem: given a set (where there are no duplicates) find all subsets of it. Say there are n items in the set. A subset either includes an item or it doesn’t, so it can be represented by a string of length n consisting of 1’s and 0’s. The original set is represented by 1, 1, ... 1 n times. The total number of subsets is 2

The program below can be modified for PBWin, Visual Basic, FORTRAN and any other BASIC-like compiler.

Global nblock As Long 'number of blocks in the original multisubset Global rep0() As Long 'representation of original multiset 'Given the representation of a multisubset, return the multisubset. Function ConvertToMultiset(rep() As Long) As String Local i, k As Long, a As String For i = 1 To nblock a = a + String$(rep(i), Chr$(97 + k)) '97 = Asc("a") Incr k Next If a = "" Then a = "{}" Function = a End Function '----------------------------------------------------------------- 'Given rep(), increment it and return whether it differs from rep0() Function IncrMultiset(rep() As Long) As Long 'rep() passed by reference Local i As Long For i = nblock To 1 Step -1 If rep(i) < rep0(i) Then Incr rep(i) Exit For Else rep(i) = 0 End If Next Function = (i >= 1) 'changed? End Function '----------------------------------------------------------------- Function Main Local rep() As Long 'representation of the current submultiset Local counter As Long 'number of submultisets Local s As String 'the original string Local n As Long 'length of s Local i, j As Long 'general indices Local u, v As Long 'general values s = "acab" 'change to suit n = Len(s) Dim item(n) As Long 's as an ascii array For i = 1 To n item(i) = Asc(s,i) Next Array Sort item(1) 'insure that repeated items are together 'Represent a b b b c c d d d for example ' by 1 3 2 3 (one A, three B's, two C's, three D's) Dim rep0(n) 'original multiset, highest possible value of nblock is n v = -1 'a non item nblock = 0 'determine number of blocks of repeated items For i = 1 To n u = item(i) counter = 0 If u = v Then Iterate 'already counted For j = 1 To n If item(j) = u Then Incr counter Next v = u Incr nblock rep0(nblock) = counter Next Print s Print ConvertToMultiset(rep0()) 's in order 'Start rep() at 0 0 0 ... 0, increment until it equals rep0() Dim rep(nblock) As Long Print "-----------------" :Print counter = 0 Do For i = 1 To nblock Print Format$(rep(i)); Next Print " "; Print ConvertToMultiset(rep()) Incr counter Loop While IncrMultiset(rep()) Print : Print "-----------------" Print "count: "; counter WaitKey$ End Function