A MultiSet is a data structure which stores and manipulates an unordered collection of elements which may be repeated.  It is implemented as a Maple object. The procedure exports of a MultiSet are used to create, update, query and otherwise interact with one or more MultiSet objects.

A MultiSet can be constructed from another MultiSet, from a set or list of elements, or from an expression sequence of elements with their multiplicities:

If M is a MultiSet, then MultiSet(M) produces a new MultiSet with the same elements and multiplicities.

If L is a Maple list or set of two-element lists, where the second element of each list is a non-negative integer (normal case) or real number (generalized case), then MultiSet(L) is a MultiSet whose elements are the first elements of each list with multiplicities given by the corresponding second elements of each list.

If L is any other Maple list, then MultiSet(L) is a MultiSet of the same elements, with multiplicities the same as in L.

If S is any other Maple set, then MultiSet(S) is a MultiSet of the same elements, each with multiplicity 1.

Each of the expressions M( a, b=2, c=3 ) and M( a, [b, 2], [c, 3] ) constructs a MultiSet in which the element a has multiplicity 1, b has multiplicity 2 and c has multiplicity 3.  Here a, b, and c can be any Maple expressions.

Multiplicities must be non-negative integers, unless the generalized index is provided on the constructor, in which case arbitrary (real) numeric multiplicities are also permitted.  An element with multiplicity 0 is removed from its MultiSet.

MultiSets are displayed using a set-of-lists-of-pairs notation, but MultiSets are not sets in the usual Maple sense.  The convert command can be used to realize a MultiSet in a variety of different alternate formats.

To test whether an expression is a MultiSet, use type(..., MultiSet).

To iterate over a MultiSet, see MultiSet Iteration.

For commands which operate on more than one MultiSet, for example intersect, at least one operand must be a MultiSet.  Other operands can be MultiSets, sets or lists; a non-MultiSet operand will be converted to a MultiSet before the operation is carried out.

Generalized and standard (non-generalized) MultiSets cannot be combined in commands which operate on more than one MultiSet.  Note that this means that if a generalized MultiSet appears in an operation, for example, union, with another argument which is not a MultiSet (for example, a list or set), then that other argument will be converted to a generalized MultiSet before proceeding with the operation.

The command IsGeneralized(M) can be used to determine if a MultiSet is generalized or not.

The following is a list of the commands which work with MultiSet objects.

$M≔\mathrm{MultiSet}\left(a=2\,\left[b\,4\right]\,c\right)$

$M≔\left\{\left[a\,2\right]\,\left[b\,4\right]\,\left[c\,1\right]\right\}$

$N≔\mathrm{MultiSet}\left[\mathrm{generalized}\right]\left(\left[\left[x\,4\right]\,\left[y\,\frac{3}{2}\right]\,\left[z\,-2\right]\right]\right)$

$N≔\left\{\left[x\,4\right]\,\left[y\,\frac{3}{2}\right]\,\left[z\,\mathrm{-2}\right]\right\}$

$\mathrm{evalb}\left(M=\mathrm{MultiSet}\left(\mathrm{Entries}\left(M\right)\right)\right)$

$\mathrm{true}$

$\mathrm{Entries}\left(M\right)$

$\left\{\left[a\,2\right]\,\left[b\,4\right]\,\left[c\,1\right]\right\}$

The MultiSet object was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.