Permutation Lists
A possible way of working with permutations is by relating them to the reorderings of the elements of a list. This is the standard point of view in the combinatorial approach to permutations, which shifts the emphasis to the permuted expressions, rather than the permutations themselves. This has always been an implicit interpretation of
permutation lists in
Mathematica, reorderings of
Range[n] for some non-negative integer
n. Several standard functions in
Mathematica allow basic manipulation of permutation lists and now other functions have been added to work with permutation lists and convert them into their disjoint cyclic form.
Basic functionality for permutation lists.
This is a list of 10 integers.
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Check that it is indeed a reordering of consecutive integers starting at 1.
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The support of the permutation list is the list of points not at their natural positions.
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If the list of integers is sorted, its support is empty.
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Permutation lists can be converted into disjoint cyclic form and vice versa. This is similar to the functions
ToCycles and
FromCycles in the
Combinatorica package in improved form.
Conversion to and from cyclic form.
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Construct the cyclic form of the permutation list. By default, singletons are removed.
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Choosing any other head keeps singletons.
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Compare with the following. Note that the cycles are returned in reversed form.
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Vice versa, you can convert a cyclic object into a permutation list of any length.
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By default the length is taken to be the largest integer present in the cycles.
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The same function allows changing the length of a permutation list without changing its support.
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Permutation lists can be used to permute the parts of an expression with the functions
Part and
Permute. The difference is that, depending on the length of the permutation list,
Part may change the number of arguments of the expression, but
Permute never changes it.
| Part | return a subexpression, possibly reordering its elements |
| Permute | permute elements of an expression as given by a permutation |
| FindPermutation | compute the permutation that takes the first list to the second |
Permuting expressions.
Take an expression and a permutation list.
Permute the elements of the expression, as indicated by the permutation list.
Part can change the number of elements.
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But
Permute never changes the length of an expression.
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Permute also accepts cyclic notation in its second argument.
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It is also possible to use a permutation group, interpreted as a set of permutations.
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If you reverse the arguments you obtain the inverse permutation.
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It is possible to perform permutation operations with permutation lists using standard commands of
Mathematica.
| Part | permutation list product |
| Ordering | permutation list inverse |
| Range | identity permutation list |
| RandomSample | pseudorandom generation of permutation lists |
Standard commands reinterpreted for permutation lists.
Take two permutation lists of the same length.
This is their product, computed with
Part.
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It corresponds to the product of permutations, if placed in the same order.
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The inverse of a permutation list can be computed with
Ordering.
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The product of the two permutation lists gives the identity permutation list.
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The identity permutation list of any length can be expressed with
Range.
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Random reorderings of the identity list are valid permutation lists.
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