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Class definitions for pseudo-lists that simplify working with structures commonly encountered in combinatorics such as permutations, combinations and subsets.

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Welcome to trotter, a 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' all selections (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 I have generalized 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).

Demos #

To see trotter in action:

  • Play Falco Shapes, a little puzzle based on Marsha Falco's game of Set.

  • Explore the huge number of permutations of letters of the alphabet with Permutation Products.

Using trotter #

Import the library:

import 'package:trotter/trotter.dart';

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 'contains' all combinations of a set of items.

Example:

    final bagOfItems = characters('abcde'),
        combos = Combinations(3, bagOfItems);
    for (final 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 'contains' all permutations of a set of items.

Example:

    final bagOfItems = characters('abcde'), perms = Permutations(3, bagOfItems);
    for (final 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(bagOfItems.length, 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 'contains' all compositions of a set of items.

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

Example:

    final bagOfItems = characters('abcde'), comps = Compositions(3, bagOfItems);
    for (final 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 'contains' all amalgams of a set of items.

Example:

    final bagOfItems = characters('abcde'), amals = Amalgams(3, bagOfItems);
    for (final 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 'contains' all subsets of a set of items.

Example:

    final bagOfItems = characters('abcde'), subs = Subsets(bagOfItems);
    for (final 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 'contains' all compounds of a set of items.

Example:

    final bagOfItems = characters('abcde'), comps = Compounds(bagOfItems);
    for (final 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:

    final largeBagOfItems = characters('abcdefghijklmnopqrstuvwxyz'),
        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! Don't 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:

    final largeBagOfItems = characters('abcdefghijklmnopqrstuvwxyz'),
        perms = Permutations(10, largeBagOfItems),
        permutationOfInterest = characters('algorithms'),
        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! (Take that, Mathematica!)

Since we sometimes can be working with indexes so large they cannot be represented using a 64 bit int, indexing and length arem implemented using BigInt.

Example:

    final largeBagOfItems = characters('abcdefghijklmnopqrstuvwxyz'),
        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 use comps().last to access the last compound in the previous example!

Syntactic sugar #

Lists #

trotter provides extensions that allow us to generate combinatoric arrangements directly from lists...

Example:

    final subsets = [1, 2, 3, 4, 5].subsets();
    for (final subset in subsets()) {
      print(subset);
    }

Output:


[]
[1]
[2]
[1, 2]
[3]
[1, 3]
[2, 3]
[1, 2, 3]
[4]
[1, 4]
[2, 4]
[1, 2, 4]
[3, 4]
[1, 3, 4]
[2, 3, 4]
[1, 2, 3, 4]
[5]
[1, 5]
[2, 5]
[1, 2, 5]
[3, 5]
[1, 3, 5]
[2, 3, 5]
[1, 2, 3, 5]
[4, 5]
[1, 4, 5]
[2, 4, 5]
[1, 2, 4, 5]
[3, 4, 5]
[1, 3, 4, 5]
[2, 3, 4, 5]
[1, 2, 3, 4, 5]

Strings #

... and strings, in which case it assumes we mean arrangements of the characters in the string.

Example:

    final subsets = 'abcde'.subsets();
    for (final subset in subsets()) {
      print(subset);
    }

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]

trotter was written by Richard Ambler.

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

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Class definitions for pseudo-lists that simplify working with structures commonly encountered in combinatorics such as permutations, combinations and subsets.

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