A combinatorics library for Node.js / Browser / XPCOM Javascript, PHP, Python, C/C++, Java

(php/python/java/c implementations in progress)

version 0.9.4 (~ 75kB minified, ~ 23kB zipped)

Abacus.js, Abacus.min.js

Abacus is a small generic library containing methods and associated math utilities for (fast) combinatorial object computation. It builds on (and extends) a deprecated previous project for PHP, Simulacra.

Abacus uses (for the most part) self-contained and standalone methods, so they can be easily copy-pasted in other projects, in case only a few methods are needed and not the whole library.

see also:

• Contemplate a fast and light-weight object-oriented Template Engine for Node.js / Browser / XPCOM Javascript, PHP, Python
• HtmlWidget html widgets used as (template) plugins and/or standalone for Node.js / Browser / XPCOM Javascript, PHP, Python both client and server-side
• Tao A simple, tiny, isomorphic, precise and fast template engine for handling both string and live dom based templates
• ModelView a light-weight and flexible MVVM framework for JavaScript/HTML5
• ModelView MVC jQueryUI Widgets plug-n-play, state-full, full-MVC widgets for jQueryUI using modelview.js (e.g calendars, datepickers, colorpickers, tables/grids, etc..) (in progress)
• Importer simple class & dependency manager and loader for Node.js / Browser / XPCOM Javascript, PHP, Python
• PublishSubscribe a simple and flexible publish-subscribe pattern implementation for Node.js / Browser / XPCOM Javascript, PHP, Python
• Dromeo a flexible, agnostic router for Node.js / Browser / XPCOM Javascript, PHP, Python
• Dialect a simple cross-vendor & cross-platform object-oriented SQL Query Builder for Node.js / Browser / XPCOM Javascript, PHP, Python
• Xpresion a simple and flexible eXpression parser engine (with custom functions and variables support) for Node.js / Browser / XPCOM Javascript, PHP, Python
• GrammarTemplate versatile and intuitive grammar-based templating for Node.js / Browser / XPCOM Javascript, PHP, Python
• GrammarPattern versatile grammar-based pattern-matching for Node.js / Browser / XPCOM Javascript (IN PROGRESS)
• Regex Analyzer/Composer Regular Expression Analyzer and Composer for Node.js / Browser / XPCOM Javascript, PHP, Python
• DateX eXtended & localised Date parsing, diffing, formatting and validation for Node.js / Browser / XPCOM Javascript, PHP, Python
• RT client-side real-time communication for Node/XPCOM/JS with support for Poll / BOSH / WebSockets
• Asynchronous a simple manager for async, linearised, parallelised, interleaved and sequential tasks for JavaScript
• Simulacra a simulation, algebraic, probability and combinatorics PHP package for scientific computations

• Live Playground Example
• Features
• Combinatorial Examples
• Performance
• Credits and References
• Example API
• Todo

Supports: (see: test/test.bat)

• Tensor (test/tensors.js)
• Tuple (test/tuples.js)
• Permutation (test/permutations.js, test/permutations-bigint.js)
• CyclicPermutation (test/cyclic_permutations.js)
• MultisetPermutation (test/multiset_permutations.js)
• DerangementPermutation (test/derangements.js) rank/unrank methods missing
• InvolutionPermutation (test/involutions.js) only counting & random generation
• ConnectedPermutation (test/connected_permutations.js) only counting & random generation
• UnorderedCombination / Combination (test/combinations.js)
• OrderedCombination / Variation / kPermutation (test/ordered_combinations.js)
• UnorderedRepeatedCombination / RepeatedCombination (test/combinations_repeats.js)
• OrderedRepeatedCombination / RepeatedVariation / kTuple (test/ordered_combinations_repeats.js)
• Subset (test/subsets.js)
• Partition (test/partitions.js) rank/unrank methods missing, partial support for COLEX
• Composition (test/compositions.js) rank/unrank methods missing, partial support for COLEX
• RestrictedPartition (test/restricted_partitions.js) exactly M max. part
• RestrictedComposition (test/restricted_compositions.js) exactly K #parts
• LatinSquare (test/latin_squares.js)
• MagicSquare (test/magic_squares.js)
• algebraic composition (of fixed dimensions at present) and sequences of combinatorial objects to construct new combinatorial objects (eg all combinations = all permutations OF all unique combinations, see test/permutations_of_combinations.js and test/permutations_of_permutations.js, k-Derangements = (n,k) Combinations combined With (n-k) Derangements, see test/k-derangements.js or all subsets = (n,0)Combinations + (n,1)Combinations + .. + (n,n-1)Combinations + (n,n)Combinations, see test/combination_subsets.js)
• custom (user-supplied callable) and/or built-in filters which can select and generate any custom and complex combinatorial object from filtering other combinatorial objects as efficiently as possible (e.g see test/filtered.js, test/filtered_partitions.js). Also algebraic / boolean composition of filters (i.e .NOT(), .AND(), .OR() and so on..). Note that filtering should be used with caution and only if no other method is currently possible to generate the desired combinatorial object as filtering is equivalent to exhaustive search over the space of the original combinatorial object and as such can be an inefficient way to generate a combinatorial object (e.g see test/filtered.js). Note2 with filtering applied some methods like .total(), .hasNext(), .base(), .dimension() still return data of the original object not the filtered object since that would require to pre-generate all the data and filter them afterwards instead of doing it one-by-one on each generation and would be impractical and unachievable for very large combinatorial objects, so be careful when using, for example, .total() with fitering applied
• multiple (combined) iterator orderings & traversals: lex, colex, random, reversed, reflected, minimal (not implemented yet). For example: "revlex" (equivalent to "lex,reversed"), "refcolex" (equivalent to "colex,reflected"), and so on..
• arbitrary range of combinatorial objects in a number of supported orderings (ie lex, colex, random,..) (and with filtering applied, if set). Note rank/unrank methods have to be implemented for this feature to work
• efficient and unbiased generation, (un)ranking, succession & random methods for supported combinatorial objects (see below)
• big-integer arithmetic, PRNGs and other math utilities can be dynamicaly pluggable using external implementations, making the lib very flexible especialy with respect to handling big-integers & (pseudo-)random number generators

• first/last, random, rank/unrank methods use efficient linear O(n) (or log-linear O(nlgn)) time and space algorithms
• random methods are statisticaly unbiased (ie uniform sampling methods, see below as well)
• successor methods use efficient CAT (ie constant average time) or Loopless (ie strictly constant time) algorithms to generate next/prev object from current object (supporting multiple combinatorial orderings along the way, see above) (note a couple of methods are linear time algorithms because the lib does not use extra space to store information between successive runs and also support static random access to successors so any extra is computed at run-time, but can easily be made CAT or even Loopless by storing extra information, eg current index position)
• avoid big-integer arithmetic and computational overhead (except if explicit ranking / unranking is needed and objects are large)

Note that the lib can generate very large (and also randomised) combinatorial objects without ever using biginteger arithmetic due to design and implementation except if arbitrary random, ranking and unranking have to be used (see above)

See the comments in the code for algorithms and references used.

A variety of combinatorial algorithms & statistics are given, for example, in:

• The Art of Computer Programming, Donald Knuth
• FXT library, Joerg Arndt and his PhD thesis
• Combinatorial Algorithms, Albert Nijenhuis, Herbert Wilf
• Combinatorial Generation, Frank Ruskey
• Generating Functionology, Herbert Wilf
• Permutation Generation Methods, Robert Sedgewick
• A Versatile Algorithm to Generate Various Combinatorial Structures, Pramod Ganapathi, Rama B
• Generating All and Random Instances of a Combinatorial Object, Ivan Stojmenovic including the section on random generation without large integers & with only one PRNG call
• Algorithms for Unranking Combinations and Other Related Choice Functions, Zbigniew Kokosinski
• Analytic Combinatorics, Philippe Flajolet, Robert Sedgewick can also be used to produce combinatorial generation algorithms instead of just generating functions for counting and statistics
• Gray Code, wikipedia
• A Survey of Combinatorial Gray Codes, Carla Savage
• Combinatorial Generation by Fusing Loopless Algorithms, Tadao Takaoka, Stephen Violich
• Make good (Pseudo-)Random Number Generators

permutations (see test/permutations.js)

Abacus.Permutations (VERSION = 0.7.5)
---
o = Abacus.Permutation(4)
o.total()
24
o.next()
[ 0, 1, 2, 3 ]
o.hasNext()
true
o.next()
[ 0, 1, 3, 2 ]
default order is "lex", lexicographic-order
o.rewind()
[ [ 0, 1, 2, 3 ],
  'index          : 0',
  'rank           : 0',
  'unrank         : 0,1,2,3',
  'is_permutation : yes',
  'is_identity    : yes',
  'is_derangement : no',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 0, 1, 3, 2 ],
  'index          : 1',
  'rank           : 1',
  'unrank         : 0,1,3,2',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 0, 2, 1, 3 ],
  'index          : 2',
  'rank           : 2',
  'unrank         : 0,2,1,3',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 0, 2, 3, 1 ],
  'index          : 3',
  'rank           : 3',
  'unrank         : 0,2,3,1',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : no' ]
[ [ 0, 3, 1, 2 ],
  'index          : 4',
  'rank           : 4',
  'unrank         : 0,3,1,2',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : no' ]
[ [ 0, 3, 2, 1 ],
  'index          : 5',
  'rank           : 5',
  'unrank         : 0,3,2,1',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 1, 0, 2, 3 ],
  'index          : 6',
  'rank           : 6',
  'unrank         : 1,0,2,3',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 1, 0, 3, 2 ],
  'index          : 7',
  'rank           : 7',
  'unrank         : 1,0,3,2',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 1, 2, 0, 3 ],
  'index          : 8',
  'rank           : 8',
  'unrank         : 1,2,0,3',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : no' ]
[ [ 1, 2, 3, 0 ],
  'index          : 9',
  'rank           : 9',
  'unrank         : 1,2,3,0',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 1, 3, 0, 2 ],
  'index          : 10',
  'rank           : 10',
  'unrank         : 1,3,0,2',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 1, 3, 2, 0 ],
  'index          : 11',
  'rank           : 11',
  'unrank         : 1,3,2,0',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 2, 0, 1, 3 ],
  'index          : 12',
  'rank           : 12',
  'unrank         : 2,0,1,3',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : no' ]
[ [ 2, 0, 3, 1 ],
  'index          : 13',
  'rank           : 13',
  'unrank         : 2,0,3,1',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 2, 1, 0, 3 ],
  'index          : 14',
  'rank           : 14',
  'unrank         : 2,1,0,3',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 2, 1, 3, 0 ],
  'index          : 15',
  'rank           : 15',
  'unrank         : 2,1,3,0',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 2, 3, 0, 1 ],
  'index          : 16',
  'rank           : 16',
  'unrank         : 2,3,0,1',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 2, 3, 1, 0 ],
  'index          : 17',
  'rank           : 17',
  'unrank         : 2,3,1,0',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : no',
  'is_connected   : no' ]
[ [ 3, 0, 1, 2 ],
  'index          : 18',
  'rank           : 18',
  'unrank         : 3,0,1,2',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 3, 0, 2, 1 ],
  'index          : 19',
  'rank           : 19',
  'unrank         : 3,0,2,1',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 3, 1, 0, 2 ],
  'index          : 20',
  'rank           : 20',
  'unrank         : 3,1,0,2',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : no',
  'is_connected   : yes' ]
[ [ 3, 1, 2, 0 ],
  'index          : 21',
  'rank           : 21',
  'unrank         : 3,1,2,0',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : no',
  'is_involution  : yes',
  'is_connected   : no' ]
[ [ 3, 2, 0, 1 ],
  'index          : 22',
  'rank           : 22',
  'unrank         : 3,2,0,1',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : no',
  'is_connected   : no' ]
[ [ 3, 2, 1, 0 ],
  'index          : 23',
  'rank           : 23',
  'unrank         : 3,2,1,0',
  'is_permutation : yes',
  'is_identity    : no',
  'is_derangement : yes',
  'is_involution  : yes',
  'is_connected   : no' ]
backwards
o.rewind(-1)
[ 3, 2, 1, 0 ]
[ 3, 2, 0, 1 ]
[ 3, 1, 2, 0 ]
[ 3, 1, 0, 2 ]
[ 3, 0, 2, 1 ]
[ 3, 0, 1, 2 ]
[ 2, 3, 1, 0 ]
[ 2, 3, 0, 1 ]
[ 2, 1, 3, 0 ]
[ 2, 1, 0, 3 ]
[ 2, 0, 3, 1 ]
[ 2, 0, 1, 3 ]
[ 1, 3, 2, 0 ]
[ 1, 3, 0, 2 ]
[ 1, 2, 3, 0 ]
[ 1, 2, 0, 3 ]
[ 1, 0, 3, 2 ]
[ 1, 0, 2, 3 ]
[ 0, 3, 2, 1 ]
[ 0, 3, 1, 2 ]
[ 0, 2, 3, 1 ]
[ 0, 2, 1, 3 ]
[ 0, 1, 3, 2 ]
[ 0, 1, 2, 3 ]
o.order("lex,reflected")
[ 3, 2, 1, 0 ]
[ 2, 3, 1, 0 ]
[ 3, 1, 2, 0 ]
[ 1, 3, 2, 0 ]
[ 2, 1, 3, 0 ]
[ 1, 2, 3, 0 ]
[ 3, 2, 0, 1 ]
[ 2, 3, 0, 1 ]
[ 3, 0, 2, 1 ]
[ 0, 3, 2, 1 ]
[ 2, 0, 3, 1 ]
[ 0, 2, 3, 1 ]
[ 3, 1, 0, 2 ]
[ 1, 3, 0, 2 ]
[ 3, 0, 1, 2 ]
[ 0, 3, 1, 2 ]
[ 1, 0, 3, 2 ]
[ 0, 1, 3, 2 ]
[ 2, 1, 0, 3 ]
[ 1, 2, 0, 3 ]
[ 2, 0, 1, 3 ]
[ 0, 2, 1, 3 ]
[ 1, 0, 2, 3 ]
[ 0, 1, 2, 3 ]
o.order("lex,reversed")
[ 3, 2, 1, 0 ]
[ 3, 2, 0, 1 ]
[ 3, 1, 2, 0 ]
[ 3, 1, 0, 2 ]
[ 3, 0, 2, 1 ]
[ 3, 0, 1, 2 ]
[ 2, 3, 1, 0 ]
[ 2, 3, 0, 1 ]
[ 2, 1, 3, 0 ]
[ 2, 1, 0, 3 ]
[ 2, 0, 3, 1 ]
[ 2, 0, 1, 3 ]
[ 1, 3, 2, 0 ]
[ 1, 3, 0, 2 ]
[ 1, 2, 3, 0 ]
[ 1, 2, 0, 3 ]
[ 1, 0, 3, 2 ]
[ 1, 0, 2, 3 ]
[ 0, 3, 2, 1 ]
[ 0, 3, 1, 2 ]
[ 0, 2, 3, 1 ]
[ 0, 2, 1, 3 ]
[ 0, 1, 3, 2 ]
[ 0, 1, 2, 3 ]
o.order("colex")
[ 3, 2, 1, 0 ]
[ 2, 3, 1, 0 ]
[ 3, 1, 2, 0 ]
[ 1, 3, 2, 0 ]
[ 2, 1, 3, 0 ]
[ 1, 2, 3, 0 ]
[ 3, 2, 0, 1 ]
[ 2, 3, 0, 1 ]
[ 3, 0, 2, 1 ]
[ 0, 3, 2, 1 ]
[ 2, 0, 3, 1 ]
[ 0, 2, 3, 1 ]
[ 3, 1, 0, 2 ]
[ 1, 3, 0, 2 ]
[ 3, 0, 1, 2 ]
[ 0, 3, 1, 2 ]
[ 1, 0, 3, 2 ]
[ 0, 1, 3, 2 ]
[ 2, 1, 0, 3 ]
[ 1, 2, 0, 3 ]
[ 2, 0, 1, 3 ]
[ 0, 2, 1, 3 ]
[ 1, 0, 2, 3 ]
[ 0, 1, 2, 3 ]
o.order("colex,reflected")
[ 0, 1, 2, 3 ]
[ 0, 1, 3, 2 ]
[ 0, 2, 1, 3 ]
[ 0, 2, 3, 1 ]
[ 0, 3, 1, 2 ]
[ 0, 3, 2, 1 ]
[ 1, 0, 2, 3 ]
[ 1, 0, 3, 2 ]
[ 1, 2, 0, 3 ]
[ 1, 2, 3, 0 ]
[ 1, 3, 0, 2 ]
[ 1, 3, 2, 0 ]
[ 2, 0, 1, 3 ]
[ 2, 0, 3, 1 ]
[ 2, 1, 0, 3 ]
[ 2, 1, 3, 0 ]
[ 2, 3, 0, 1 ]
[ 2, 3, 1, 0 ]
[ 3, 0, 1, 2 ]
[ 3, 0, 2, 1 ]
[ 3, 1, 0, 2 ]
[ 3, 1, 2, 0 ]
[ 3, 2, 0, 1 ]
[ 3, 2, 1, 0 ]
o.order("colex,reversed")
[ 0, 1, 2, 3 ]
[ 1, 0, 2, 3 ]
[ 0, 2, 1, 3 ]
[ 2, 0, 1, 3 ]
[ 1, 2, 0, 3 ]
[ 2, 1, 0, 3 ]
[ 0, 1, 3, 2 ]
[ 1, 0, 3, 2 ]
[ 0, 3, 1, 2 ]
[ 3, 0, 1, 2 ]
[ 1, 3, 0, 2 ]
[ 3, 1, 0, 2 ]
[ 0, 2, 3, 1 ]
[ 2, 0, 3, 1 ]
[ 0, 3, 2, 1 ]
[ 3, 0, 2, 1 ]
[ 2, 3, 0, 1 ]
[ 3, 2, 0, 1 ]
[ 1, 2, 3, 0 ]
[ 2, 1, 3, 0 ]
[ 1, 3, 2, 0 ]
[ 3, 1, 2, 0 ]
[ 2, 3, 1, 0 ]
[ 3, 2, 1, 0 ]
o.order("random")
[ 1, 3, 0, 2 ]
[ 0, 2, 3, 1 ]
[ 2, 3, 1, 0 ]
[ 0, 2, 1, 3 ]
[ 0, 1, 3, 2 ]
[ 1, 2, 0, 3 ]
[ 0, 3, 2, 1 ]
[ 2, 3, 0, 1 ]
[ 0, 1, 2, 3 ]
[ 1, 0, 2, 3 ]
[ 3, 2, 1, 0 ]
[ 1, 0, 3, 2 ]
[ 3, 1, 0, 2 ]
[ 2, 0, 1, 3 ]
[ 1, 2, 3, 0 ]
[ 1, 3, 2, 0 ]
[ 3, 0, 2, 1 ]
[ 2, 0, 3, 1 ]
[ 3, 1, 2, 0 ]
[ 2, 1, 3, 0 ]
[ 0, 3, 1, 2 ]
[ 3, 2, 0, 1 ]
[ 3, 0, 1, 2 ]
[ 2, 1, 0, 3 ]
o.random()
[ 0, 1, 3, 2 ]
o.order("colex").range(-5, -1)
[ 1, 2, 0, 3 ]
[ 2, 0, 1, 3 ]
[ 0, 2, 1, 3 ]
[ 1, 0, 2, 3 ]
[ 0, 1, 2, 3 ]
o.dispose()

big-integer arithmetic permutations (see test/permutations-bigint.js)

Abacus.Permutations (VERSION = 0.7.6)
---
o = Abacus.Permutation(50)
o.total()
30414093201713378043612608166064768844377641568960512000000000000
o.random()
21,14,4,48,13,45,43,42,44,38,2,32,7,15,3,30,46,29,24,18,23,19,47,39,12,6,11,37,1,20,16,5,9,36,8,22,35,49,34,28,31,26,10,27,25,0,41,40,33,17
o.item(78043612608166064768844377641568960512000000000000,"lex")
0,1,2,3,4,5,6,7,10,22,36,11,30,34,12,27,9,31,26,20,48,19,18,47,13,24,14,21,17,38,16,15,41,40,43,23,28,39,46,37,35,45,8,33,42,29,44,49,25,32
o.item(78043612608166064768844377641568960512000000000000,"colex")
32,25,49,44,29,42,33,8,45,35,37,46,39,28,23,43,40,41,15,16,38,17,21,14,24,13,47,18,19,48,20,26,31,9,27,12,34,30,11,36,22,10,7,6,5,4,3,2,1,0
o.item(78043612608166064768844377641568960512000000000000,"revlex")
49,48,47,46,45,44,43,42,39,27,13,38,19,15,37,22,40,18,23,29,1,30,31,2,36,25,35,28,32,11,33,34,8,9,6,26,21,10,3,12,14,4,41,16,7,20,5,0,24,17
o.item(78043612608166064768844377641568960512000000000000,"revcolex")
17,24,0,5,20,7,16,41,4,14,12,3,10,21,26,6,9,8,34,33,11,32,28,35,25,36,2,31,30,1,29,23,18,40,22,37,15,19,38,13,27,39,42,43,44,45,46,47,48,49
o.order("lex").range(30414093201713378043612608166064768844377641568960511999999999998)
49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,0,1
49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0
o.dispose()

• apply built-in language iterator/iterable patterns (e.g ES6 iterator protocol, Python __iter__ interface, PHP Iterator interface, ..). Combinatorial objects additionaly support a doubly-linked list-like interface, i.e prev/next accessors [DONE]
• support biginteger combinatorial computations e.g large factorials [DONE], the lib does not support biginteger arithmetic, but arithmetic routines have been made dynamicaly pluggable and one can use an external implementation to support combinatorics with bigintegers where needed as needed, see test examples for an example
• support multiple combined custom iterator orderings, i.e lex, colex, reversed, reflected, random seamlessly and uniformly, both forward and backward [DONE, random ordering may be optimised further]
• support efficient successor methods (preferably CAT/Loopless methods) to generate next/prev object from current object [DONE]
• support efficient ranking / unranking algorithms and associated methods (preferably of O(n) or O(nlgn) complexity) for supported orderings [DONE]
• support multiple combinatorial orderings (ie lex, colex, reflex, refcolex, minimal, ..) directly in the successor methods instead of using post-transformations on object [DONE]
• support unique and uniform random ordering traversals for all combinatorial objects, so that the space of a combinatorial object can be traversed in any random ordering uniquely and unbiasedly (useful in some applications, eg backtracking) [DONE, see reference, used as custom iterator ordering, see above, may be optimised further]
• make sure the .random methods uniformly and unbiasedly sample the combinatorial object space (methods use unbiased sampling algorithms, however results in certain cases might depend on quality of PRNGs) [DONE]
• support algebraic composition/cascading of combinatorial objects (of fixed dimensions at present) to construct new combinatorial objects (eg all combinations = all permutations OF all unique combinations) [DONE]
• support generation of supported combinatorial objects with additional user-defined patterns/templates of constraints to satisfy e.g "only combinatorial objects matching '(n)(m)(1){2}(){3}(0)((n+1))((n+m)){4}'" pattern.. [DONE]
• add LatinSquare, MagicSquare algorithms [DONE]
• add run-time/lazy custom and/or built-in filtering support (with support for filter composition as well) to generate and select custom and complex combinatorial objects from filtering other combinatorial objects as efficiently as possible [DONE]
• add efficient rank/unrank methods for DerangementPermutation, InvolutionPermutation, ConnectedPermutation, Composition & Partition (TODO)
• full support for colex ordering Composition & Partition [DONE PARTIALY]
• support minimal/gray ordering (and successor) for all supported combinatorial objects (TODO)
• use numeric arrays (ie Uint32) to store combinatorial items and/or make faster successor methods and other numerical routines to asm.js (TODO?)
• support generation (and counting) of combinatorial objects (including the basic supported ones) based on generic user-defined symbolic constraints / symmetries / rules to satisfy, for example permutations defined symbolicaly and directly by their symmetries / constraints instead of being hardcoded as elementary objects (TODO?, see using filtering as a similar alternative to this approach)
• support graph-based combinatorial objects like Graph, Grammar,.. (TODO?) (for regular grammars and expressions see RegexAnalyzer for an example)