Welcome to trotter, a Dart library that simplifies working with meta-arrangements commonly encountered in combinatorics such as arrangements of combinations and permutations.

trotter gives the developer access to pseudo-lists that "contain" an arrangement of all arrangements (combinations, permutations, etc.) of objects taken from a specified list of items.

The order of arrangements is based on the the order produced by the Steinhaus–Johnson–Trotter algorithm for ordering permutations, which has been generalized in this library to combinations and arrangements that allow for replacement after item selection.

The pseudo-list classes available are:

  • Combinations.
  • Permutations.
  • Compositions (combinations with replacement).
  • Amalgams (permutations with replacement).
  • Subsets (combinations of unspecified size).
  • Compounds (permutations of unspecified size).

Demo

Take a look at Falco Shapes, a trotter demo based on Marsha Falco's combinatorics game Set, that uses trotter to search through items for combinations with certain characteristics.

The basic classes

Combinations

A combination is a selection of items for which order is not important and items are not replaced after being selected.

The Combinations class defines a pseudo-list that "contains" an arrangement of all combinations of a set of items.

Example:

    List bagOfItems = characters("abcde");
    var combos = Combinations(3, bagOfItems);
    for (var combo in combos()) {
      print(combo);
    }

Output:


[a, b, c]
[a, b, d]
[a, b, e]
[a, c, d]
[a, c, e]
[a, d, e]
[b, c, d]
[b, c, e]
[b, d, e]
[c, d, e]

Permutations

A permutation is a selection of items for which order is important and items are not replaced after being selected.

The Permutations class defines a pseudo-list that "contains" an arrangement of all permutations of a set of items.

Example:

    List bagOfItems = characters("abcde");
    var perms = Permutations(3, bagOfItems);
    for (var perm in perms()) {
      print(perm);
    }

Output:


[a, b, c]
[a, c, b]
[c, a, b]
[c, b, a]
[b, c, a]
[b, a, c]
[a, b, d]
[a, d, b]
[d, a, b]
[d, b, a]
[b, d, a]
[b, a, d]
[a, b, e]
[a, e, b]
[e, a, b]
[e, b, a]
[b, e, a]
[b, a, e]
[a, c, d]
[a, d, c]
[d, a, c]
[d, c, a]
[c, d, a]
[c, a, d]
[a, c, e]
[a, e, c]
[e, a, c]
[e, c, a]
[c, e, a]
[c, a, e]
[a, d, e]
[a, e, d]
[e, a, d]
[e, d, a]
[d, e, a]
[d, a, e]
[b, c, d]
[b, d, c]
[d, b, c]
[d, c, b]
[c, d, b]
[c, b, d]
[b, c, e]
[b, e, c]
[e, b, c]
[e, c, b]
[c, e, b]
[c, b, e]
[b, d, e]
[b, e, d]
[e, b, d]
[e, d, b]
[d, e, b]
[d, b, e]
[c, d, e]
[c, e, d]
[e, c, d]
[e, d, c]
[d, e, c]
[d, c, e]

(Notice that this library arranges permutations similarly to the way the Steinhaus-Johnson-Trotter algorithm arranges permutations. In fact, if we get the permutations of all the specified items, e.g. var perms = Permutations(5, bagOfItems); in the above code, the arrangement of permutations is exactly what would have resulted from applying the S-J-T algorithm. The algorithms in this library have an advantage in that they do not iterate through all k - 1 permutations in order to determint the kth permutation, however.)

Compositions

A composition (or combination with replacement) is a selection of items for which order is not important and items are replaced after being selected.

The Compositions class defines a pseudo-list that "contains" an arrangement of all compositions of a set of items.

Here are all the compositions of three items taken from a bag of five items:

Example:

    List bagOfItems = characters("abcde");
    var comps = Compositions(3, bagOfItems);
    for (var comp in comps()) {
      print(comp);
    }

Output:


[a, a, a]
[a, a, b]
[a, a, c]
[a, a, d]
[a, a, e]
[a, b, b]
[a, b, c]
[a, b, d]
[a, b, e]
[a, c, c]
[a, c, d]
[a, c, e]
[a, d, d]
[a, d, e]
[a, e, e]
[b, b, b]
[b, b, c]
[b, b, d]
[b, b, e]
[b, c, c]
[b, c, d]
[b, c, e]
[b, d, d]
[b, d, e]
[b, e, e]
[c, c, c]
[c, c, d]
[c, c, e]
[c, d, d]
[c, d, e]
[c, e, e]
[d, d, d]
[d, d, e]
[d, e, e]
[e, e, e]

Amalgams

An amalgam (or permutation with replacement) is a selection of items for which order is important and items are replaced after being selected.

The Amalgams class defines a pseudo-list that "contains" an arrangement of all amalgams of a set of items.

Example:

    List bagOfItems = characters("abcde");
    var amals = Amalgams(3, bagOfItems);
    for (var amal in amals()) {
      print(amal);
    }

Output:


[a, a, a]
[a, a, b]
[a, a, c]
[a, a, d]
[a, a, e]
[a, b, a]
[a, b, b]
[a, b, c]
[a, b, d]
[a, b, e]
[a, c, a]
[a, c, b]
[a, c, c]
[a, c, d]
[a, c, e]
[a, d, a]
[a, d, b]
[a, d, c]
[a, d, d]
[a, d, e]
[a, e, a]
[a, e, b]
[a, e, c]
[a, e, d]
[a, e, e]
[b, a, a]
[b, a, b]
[b, a, c]
[b, a, d]
[b, a, e]
[b, b, a]
[b, b, b]
[b, b, c]
[b, b, d]
[b, b, e]
[b, c, a]
[b, c, b]
[b, c, c]
[b, c, d]
[b, c, e]
[b, d, a]
[b, d, b]
[b, d, c]
[b, d, d]
[b, d, e]
[b, e, a]
[b, e, b]
[b, e, c]
[b, e, d]
[b, e, e]
[c, a, a]
[c, a, b]
[c, a, c]
[c, a, d]
[c, a, e]
[c, b, a]
[c, b, b]
[c, b, c]
[c, b, d]
[c, b, e]
[c, c, a]
[c, c, b]
[c, c, c]
[c, c, d]
[c, c, e]
[c, d, a]
[c, d, b]
[c, d, c]
[c, d, d]
[c, d, e]
[c, e, a]
[c, e, b]
[c, e, c]
[c, e, d]
[c, e, e]
[d, a, a]
[d, a, b]
[d, a, c]
[d, a, d]
[d, a, e]
[d, b, a]
[d, b, b]
[d, b, c]
[d, b, d]
[d, b, e]
[d, c, a]
[d, c, b]
[d, c, c]
[d, c, d]
[d, c, e]
[d, d, a]
[d, d, b]
[d, d, c]
[d, d, d]
[d, d, e]
[d, e, a]
[d, e, b]
[d, e, c]
[d, e, d]
[d, e, e]
[e, a, a]
[e, a, b]
[e, a, c]
[e, a, d]
[e, a, e]
[e, b, a]
[e, b, b]
[e, b, c]
[e, b, d]
[e, b, e]
[e, c, a]
[e, c, b]
[e, c, c]
[e, c, d]
[e, c, e]
[e, d, a]
[e, d, b]
[e, d, c]
[e, d, d]
[e, d, e]
[e, e, a]
[e, e, b]
[e, e, c]
[e, e, d]
[e, e, e]

Subsets

A subset (or combination of unspecified length) is a selection of items for which order is not important, items are not replaced and the number of items is not specified.

The Subsets class defines a pseudo-list that "contains" an arrangement of all subsets of a set of items.

Example:

    List bagOfItems = characters("abcde");
    var subs = Subsets(bagOfItems);
    for (var sub in subs()) {
      print(sub);
    }

Output:


[]
[a]
[b]
[a, b]
[c]
[a, c]
[b, c]
[a, b, c]
[d]
[a, d]
[b, d]
[a, b, d]
[c, d]
[a, c, d]
[b, c, d]
[a, b, c, d]
[e]
[a, e]
[b, e]
[a, b, e]
[c, e]
[a, c, e]
[b, c, e]
[a, b, c, e]
[d, e]
[a, d, e]
[b, d, e]
[a, b, d, e]
[c, d, e]
[a, c, d, e]
[b, c, d, e]
[a, b, c, d, e]

Compounds

A compound (or permutation of unspecified length) is a selection of items for which order is important, items are not replaced and the number of items is not specified.

The Compounds class defines a pseudo-list that "contains" an arrangement of all compounds of a set of items.

Example:

    List bagOfItems = characters("abcde");
    var comps = Compounds(bagOfItems);
    for (var comp in comps()) {
      print(comp);
    }

Output:


[]
[a]
[b]
[c]
[d]
[e]
[a, b]
[b, a]
[a, c]
[c, a]
[a, d]
[d, a]
[a, e]
[e, a]
[b, c]
[c, b]
[b, d]
[d, b]
[b, e]
[e, b]
[c, d]
[d, c]
[c, e]
[e, c]
[d, e]
[e, d]
[a, b, c]
[a, c, b]
[c, a, b]
[c, b, a]
[b, c, a]
[b, a, c]
[a, b, d]
[a, d, b]
[d, a, b]
[d, b, a]
[b, d, a]
[b, a, d]
[a, b, e]
[a, e, b]
[e, a, b]
[e, b, a]
[b, e, a]
[b, a, e]
[a, c, d]
[a, d, c]
[d, a, c]
[d, c, a]
[c, d, a]
[c, a, d]
[a, c, e]
[a, e, c]
[e, a, c]
[e, c, a]
[c, e, a]
[c, a, e]
[a, d, e]
[a, e, d]
[e, a, d]
[e, d, a]
[d, e, a]
[d, a, e]
[b, c, d]
[b, d, c]
[d, b, c]
[d, c, b]
[c, d, b]
[c, b, d]
[b, c, e]
[b, e, c]
[e, b, c]
[e, c, b]
[c, e, b]
[c, b, e]
[b, d, e]
[b, e, d]
[e, b, d]
[e, d, b]
[d, e, b]
[d, b, e]
[c, d, e]
[c, e, d]
[e, c, d]
[e, d, c]
[d, e, c]
[d, c, e]
[a, b, c, d]
[a, b, d, c]
[a, d, b, c]
[d, a, b, c]
[d, a, c, b]
[a, d, c, b]
[a, c, d, b]
[a, c, b, d]
[c, a, b, d]
[c, a, d, b]
[c, d, a, b]
[d, c, a, b]
[d, c, b, a]
[c, d, b, a]
[c, b, d, a]
[c, b, a, d]
[b, c, a, d]
[b, c, d, a]
[b, d, c, a]
[d, b, c, a]
[d, b, a, c]
[b, d, a, c]
[b, a, d, c]
[b, a, c, d]
[a, b, c, e]
[a, b, e, c]
[a, e, b, c]
[e, a, b, c]
[e, a, c, b]
[a, e, c, b]
[a, c, e, b]
[a, c, b, e]
[c, a, b, e]
[c, a, e, b]
[c, e, a, b]
[e, c, a, b]
[e, c, b, a]
[c, e, b, a]
[c, b, e, a]
[c, b, a, e]
[b, c, a, e]
[b, c, e, a]
[b, e, c, a]
[e, b, c, a]
[e, b, a, c]
[b, e, a, c]
[b, a, e, c]
[b, a, c, e]
[a, b, d, e]
[a, b, e, d]
[a, e, b, d]
[e, a, b, d]
[e, a, d, b]
[a, e, d, b]
[a, d, e, b]
[a, d, b, e]
[d, a, b, e]
[d, a, e, b]
[d, e, a, b]
[e, d, a, b]
[e, d, b, a]
[d, e, b, a]
[d, b, e, a]
[d, b, a, e]
[b, d, a, e]
[b, d, e, a]
[b, e, d, a]
[e, b, d, a]
[e, b, a, d]
[b, e, a, d]
[b, a, e, d]
[b, a, d, e]
[a, c, d, e]
[a, c, e, d]
[a, e, c, d]
[e, a, c, d]
[e, a, d, c]
[a, e, d, c]
[a, d, e, c]
[a, d, c, e]
[d, a, c, e]
[d, a, e, c]
[d, e, a, c]
[e, d, a, c]
[e, d, c, a]
[d, e, c, a]
[d, c, e, a]
[d, c, a, e]
[c, d, a, e]
[c, d, e, a]
[c, e, d, a]
[e, c, d, a]
[e, c, a, d]
[c, e, a, d]
[c, a, e, d]
[c, a, d, e]
[b, c, d, e]
[b, c, e, d]
[b, e, c, d]
[e, b, c, d]
[e, b, d, c]
[b, e, d, c]
[b, d, e, c]
[b, d, c, e]
[d, b, c, e]
[d, b, e, c]
[d, e, b, c]
[e, d, b, c]
[e, d, c, b]
[d, e, c, b]
[d, c, e, b]
[d, c, b, e]
[c, d, b, e]
[c, d, e, b]
[c, e, d, b]
[e, c, d, b]
[e, c, b, d]
[c, e, b, d]
[c, b, e, d]
[c, b, d, e]
[a, b, c, d, e]
[a, b, c, e, d]
[a, b, e, c, d]
[a, e, b, c, d]
[e, a, b, c, d]
[e, a, b, d, c]
[a, e, b, d, c]
[a, b, e, d, c]
[a, b, d, e, c]
[a, b, d, c, e]
[a, d, b, c, e]
[a, d, b, e, c]
[a, d, e, b, c]
[a, e, d, b, c]
[e, a, d, b, c]
[e, d, a, b, c]
[d, e, a, b, c]
[d, a, e, b, c]
[d, a, b, e, c]
[d, a, b, c, e]
[d, a, c, b, e]
[d, a, c, e, b]
[d, a, e, c, b]
[d, e, a, c, b]
[e, d, a, c, b]
[e, a, d, c, b]
[a, e, d, c, b]
[a, d, e, c, b]
[a, d, c, e, b]
[a, d, c, b, e]
[a, c, d, b, e]
[a, c, d, e, b]
[a, c, e, d, b]
[a, e, c, d, b]
[e, a, c, d, b]
[e, a, c, b, d]
[a, e, c, b, d]
[a, c, e, b, d]
[a, c, b, e, d]
[a, c, b, d, e]
[c, a, b, d, e]
[c, a, b, e, d]
[c, a, e, b, d]
[c, e, a, b, d]
[e, c, a, b, d]
[e, c, a, d, b]
[c, e, a, d, b]
[c, a, e, d, b]
[c, a, d, e, b]
[c, a, d, b, e]
[c, d, a, b, e]
[c, d, a, e, b]
[c, d, e, a, b]
[c, e, d, a, b]
[e, c, d, a, b]
[e, d, c, a, b]
[d, e, c, a, b]
[d, c, e, a, b]
[d, c, a, e, b]
[d, c, a, b, e]
[d, c, b, a, e]
[d, c, b, e, a]
[d, c, e, b, a]
[d, e, c, b, a]
[e, d, c, b, a]
[e, c, d, b, a]
[c, e, d, b, a]
[c, d, e, b, a]
[c, d, b, e, a]
[c, d, b, a, e]
[c, b, d, a, e]
[c, b, d, e, a]
[c, b, e, d, a]
[c, e, b, d, a]
[e, c, b, d, a]
[e, c, b, a, d]
[c, e, b, a, d]
[c, b, e, a, d]
[c, b, a, e, d]
[c, b, a, d, e]
[b, c, a, d, e]
[b, c, a, e, d]
[b, c, e, a, d]
[b, e, c, a, d]
[e, b, c, a, d]
[e, b, c, d, a]
[b, e, c, d, a]
[b, c, e, d, a]
[b, c, d, e, a]
[b, c, d, a, e]
[b, d, c, a, e]
[b, d, c, e, a]
[b, d, e, c, a]
[b, e, d, c, a]
[e, b, d, c, a]
[e, d, b, c, a]
[d, e, b, c, a]
[d, b, e, c, a]
[d, b, c, e, a]
[d, b, c, a, e]
[d, b, a, c, e]
[d, b, a, e, c]
[d, b, e, a, c]
[d, e, b, a, c]
[e, d, b, a, c]
[e, b, d, a, c]
[b, e, d, a, c]
[b, d, e, a, c]
[b, d, a, e, c]
[b, d, a, c, e]
[b, a, d, c, e]
[b, a, d, e, c]
[b, a, e, d, c]
[b, e, a, d, c]
[e, b, a, d, c]
[e, b, a, c, d]
[b, e, a, c, d]
[b, a, e, c, d]
[b, a, c, e, d]
[b, a, c, d, e]

Large indices

Arrangement numbers often grow very quickly. For example, consider the number of 10-permutations of the letters of the alphabet:

Example:

    List largeBagOfItems = characters("abcdefghijklmnopqrstuvwxyz");
    var perms = Permutations(10, largeBagOfItems);
    print(perms);

Output:


Pseudo-list containing all 19275223968000 10-permutations of items from [a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z].

Wow! That's a lot of permutations! It's most likely a bad idea to iterate over them all!

Notice that the word algorithms is a 10-permutation of the letters of the alphabet. What is the index of this permutation in our list of permutations?

Example:

    List largeBagOfItems = characters("abcdefghijklmnopqrstuvwxyz");
    var perms = Permutations(10, largeBagOfItems);
    List permutationOfInterest = characters("algorithms");
    BigInt index = perms.indexOf(permutationOfInterest);
    print("The index of $permutationOfInterest is $index.");
    print("perms[$index]: ${perms[index]}");

Output:


The index of [a, l, g, o, r, i, t, h, m, s] is 6831894769563.
perms[6831894769563]: [a, l, g, o, r, i, t, h, m, s]

Wow! That's almost seven trillion! Luckily we didn't need to perform that search using brute force!

Be aware that we sometimes can be working with indexes so large that they cannot be represented using Dart's 64 bit int, in which case we need to use BigInt objects.

Example:

    var largeBagOfItems = characters("abcdefghijklmnopqrstuvwxyz");
    var comps = Compounds(largeBagOfItems);
    print("There are ${comps.length} compounds of these letters!");
    BigInt lastCompoundIndex = comps.length - BigInt.one;
    print("The last compound is ${comps[lastCompoundIndex]}.");

Output:


There are 1096259850353149530222034277 compounds of these letters!
The last compound is [b, a, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z].

Unless you're immortal, don't try to use comps().last to access the last compound in the previous example!

trotter in Dart 2

In Dart 1, at least on the Dart VM, int instances could be used to represent arbitrary precision integers, and the classes above could conveniently extend ListBase, which made the analogy with a list of arrangements very strong. As of Dart 2, int instances can only be used to represent up to 64 bit integers. Although Dart 2 does provide the BigInt class for dealing with very large integers, BigInt instances cannot be used to index List instances, and the classes above had to drop the extension.

The first version of trotter that can be used for large structures in Dart 2 is trotter 0.9.5. In general, only slight modifications need be made to code written for previous versions. Here are some examples.

Instances are no longer iterable

Instances are no longer iterable, list-like structures. An Iterable "containing" all the arrangements is available through directly calling the instance, calling the range method or accessing the iterable property, however.

trotter < 0.9.5

var combos = Combinations(3, characters("abcde"));
for (var combo in combos) { // combos is iterable
...
}

trotter >= 0.9.5

var combos = Combinations(3, characters("abcde"));
for (var combo in combos()) { // combos is callable, returns an iterable
...
}

Filters, mappings and other tasks associated with Iterable instances can no longer be applied directly to instances of the classes above, but can, of course be applied to the Iterable returned by direct calling, the range method or the iterable property.

Example:

    var items = characters("abc");
    var subsets = Subsets(items);
    print(subsets().where((subset) => subset.length == 2).join(" "));

Output:


[a, b] [a, c] [b, c]

The Selections class has been renamed

In combinatorics literature the term selection is often used as a generic word that can refer to permutations or combinations in different contexts. The use of the term for combinations with replacement could thus be confusing. As of trotter 0.9.5, the class Compositions is used to represent combinations. I feel that composition is a fitting word: if a body is composed of items A, B and C then it is also composed of C, B and A, so composition suggests that order is not important. Further a body can be composed of two parts of A to one part of B, which suggests that items are replaced after being selected.

trotter was written by Richard Ambler.

Thanks for your interest in this library. Please file any bugs, issues and suggestions here.

Libraries

trotter
Classes for representing structures commonly encountered in combinatorics. [...]