Efficient combinatorial algorithms over multisets, including
generating all permutations, partitions, subsets, cycles, and
other combinatorial structures based on multisets. Note that an
`Eq`

or `Ord`

instance on the elements is *not* required; the
algorithms are careful to keep track of which things are (by
construction) equal to which other things, so equality testing is
not needed.

- type Count = Int
- newtype Multiset a = MS {}
- consMS :: (a, Count) -> Multiset a -> Multiset a
- (+:) :: (a, Count) -> Multiset a -> Multiset a
- toList :: Multiset a -> [a]
- fromList :: Ord a => [a] -> Multiset a
- fromListEq :: Eq a => [a] -> Multiset a
- fromDistinctList :: [a] -> Multiset a
- fromCounts :: [(a, Count)] -> Multiset a
- getCounts :: Multiset a -> [Count]
- disjUnion :: Multiset a -> Multiset a -> Multiset a
- disjUnions :: [Multiset a] -> Multiset a
- permutations :: Multiset a -> [[a]]
- permutationsRLE :: Multiset a -> [[(a, Count)]]
- type Vec = [Count]
- vPartitions :: Vec -> [Multiset Vec]
- partitions :: Multiset a -> [Multiset (Multiset a)]
- splits :: Multiset a -> [(Multiset a, Multiset a)]
- kSubsets :: Count -> Multiset a -> [Multiset a]
- cycles :: Multiset a -> [[a]]
- sequenceMS :: Multiset [a] -> [Multiset a]

# The `Multiset`

type

A multiset is represented as a list of (element, count) pairs. We maintain the invariants that the counts are always positive, and no element ever appears more than once.

consMS :: (a, Count) -> Multiset a -> Multiset aSource

Add an element with multiplicity to a multiset. Precondition: the new element is distinct from all elements already in the multiset.

## Conversions

fromList :: Ord a => [a] -> Multiset aSource

Efficiently convert a list to a multiset, given an `Ord`

instance
for the elements. This method is provided just for convenience.
you can also use `fromListEq`

with only an `Eq`

instance, or
construct `Multiset`

s directly using `fromCounts`

.

fromListEq :: Eq a => [a] -> Multiset aSource

Convert a list to a multiset, given an `Eq`

instance for the
elements.

fromDistinctList :: [a] -> Multiset aSource

Make a multiset with one copy of each element from a list of distinct elements.

fromCounts :: [(a, Count)] -> Multiset aSource

Construct a `Multiset`

from a list of (element, count) pairs.
Precondition: the counts must all be positive, and there must not
be any duplicate elements.

getCounts :: Multiset a -> [Count]Source

Extract just the element counts from a multiset, forgetting the elements.

## Operations

disjUnion :: Multiset a -> Multiset a -> Multiset aSource

Form the disjoint union of two multisets; i.e. we assume the two multisets share no elements in common.

disjUnions :: [Multiset a] -> Multiset aSource

Form the disjoint union of a collection of multisets. We assume that the multisets all have distinct elements.

# Permutations

permutations :: Multiset a -> [[a]]Source

List all the distinct permutations of the elements of a multiset.

For example, ```
permutations (fromList "abb") ==
["abb","bba","bab"]
```

, whereas ```
Data.List.permutations
"abb" == ["abb","bab","bba","bba","bab","abb"]
```

.
This function is equivalent to, but *much* more efficient than,
`nub . Data.List.permutations`

, and even works when the elements
have no `Eq`

instance.

Note that this is a specialized version of `permutationsRLE`

,
where each run has been expanded via `replicate`

.

permutationsRLE :: Multiset a -> [[(a, Count)]]Source

List all the distinct permutations of the elements of a multiset, with each permutation run-length encoded. (Note that the run-length encoding is a natural byproduct of the algorithm used, not a separate postprocessing step.)

For example, `permutationsRLE [(`

.
`a`

,1), (`b`

,2)] ==
[[(`a`

,1),(`b`

,2)],[(`b`

,2),(`a`

,1)],[(`b`

,1),(`a`

,1),(`b`

,1)]]

(Note that although the output type is newtype-equivalent to
`[Multiset a]`

, we don't call it that since the output may
violate the `Multiset`

invariant that no element should appear
more than once. And indeed, morally this function does not
output multisets at all.)

# Partitions

vPartitions :: Vec -> [Multiset Vec]Source

Generate all vector partitions, representing each partition as a multiset of vectors.

This code is a slight generalization of the code published in

Brent Yorgey. "Generating Multiset Partitions". In: The Monad.Reader, Issue 8, September 2007. http://www.haskell.org/sitewiki/images/d/dd/TMR-Issue8.pdf

See that article for a detailed discussion of the code and how it works.

partitions :: Multiset a -> [Multiset (Multiset a)]Source

Efficiently generate all distinct multiset partitions. Note that each partition is represented as a multiset of parts (each of which is a multiset) in order to properly reflect the fact that some parts may occur multiple times.

# Submultisets

splits :: Multiset a -> [(Multiset a, Multiset a)]Source

Generate all splittings of a multiset into two submultisets, i.e. all size-two partitions.

# Cycles

cycles :: Multiset a -> [[a]]Source

Generate all distinct cycles, aka necklaces, with elements taken from a multiset. See J. Sawada, "A fast algorithm to generate necklaces with fixed content", J. Theor. Comput. Sci. 301 (2003) pp. 477-489.

Given the ordering on the elements of the multiset based on their
position in the multiset representation (with "smaller"
elements first), in `map reverse (cycles m)`

, each generated
cycle is lexicographically smallest among all its cyclic shifts,
and furthermore, the cycles occur in reverse lexicographic
order. (It's simply more convenient/efficient to generate the
cycles reversed in this way, and of course we get the same set of
cycles either way.)

For example, ```
cycles (fromList "aabbc") ==
["cabba","bcaba","cbaba","bbcaa","bcbaa","cbbaa"]
```

.

# Miscellaneous

sequenceMS :: Multiset [a] -> [Multiset a]Source

Take a multiset of lists, and select one element from each list in every possible combination to form a list of multisets. We assume that all the list elements are distinct.