PeterBurbery/Combinatorics

(2.0.16) current version: 2.1.1 »

Combinatorics functions for subsets, tuples, and permutations

Contributed by: Peter Cullen Burbery

This paclet is applications of combinatorics. I am a combinatorialist.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["PeterBurbery/Combinatorics"]

To load the code after installation, evaluate this code:
Needs["PeterBurbery`Combinatorics`"]

Details

I would like to give credit to Ed Pegg Jr who developed many of the functions that make up this paclet. Many of the functions are available from the Resource Function Repository.

Not all the functions in this paclet are related to combinatorics. Some of the functions, like RationalNumberRepeatingDecimalPeriod are for Number Theory. I might create a paclet for Number Theory in the future, but for now, I'm just going to put all the functions in this paclet.

I made sure to provide attribution for functions bsaed on Resource Functions that other people created.

This paclet is set to load on startup.

To load the tutorial/technote Combinatorics, type PeterBurbery/Combinatorics/tutorial/Combinatorics into the Documentation Center's address bar.

Paclet Guide

Some of the functions here do not belong in combinatorics. I am including them here anyways.

Indexing

PermutationIndex

— Give the lexicographic index of a permutation

PermutationFromIndex

— Give the permutation corresponding to a given length and lexicographic index

SubsetIndex

— Give the index of a subset

SubsetFromIndex

— Get the subset with the given index and length

TupleIndex

— Compute the index of a given tuple of non-negative integers

TupleFromIndex

— Return the tuple corresponding to a given index

OrderedTupleIndex

— Give the index of an ordered tuple

OrderedTupleFromIndex

— Get the ordered tuple with the given index and length

LehmerCodeFromPermutation

— Create the Lehmer code corresponding to a given permutation

Partitions

ConjugatePartition

— Switch the rows and columns of a partition

DominatingIntegerPartitionQ

— Determine if one integer partition dominates another

IntegerPartitionQ

— Check whether the argument is a weakly decreasing list of positive integers

PartitionCrank

— Get Dyson's crank of an integer partition

PartitionFromFrobeniusSymbol

— Construct the corresponding partition, given a Frobenius symbol

PartitionRank

— The largest part minus the number of parts of a partition

FrobeniusSymbolFromPartition

— Get a pair of lists representing the Ferrers diagram of a partition

FerrersDiagram

— Display the Ferrers diagram of a partition with dots

DurfeeSquare

— get the dimensions of the Durfee square of a partition

SelfConjugatePartitionQ

— test if a partition is a self-conjugate partition

Tableaux

TableauQ

— Determine if a list of values forms a Young tableau

TransposeTableau

— Transpose a Young tableau

PermutationToTableaux

— Get the two Young tableaux corresponding to a permutation

TableauxToPermutation

— Get a permutation corresponding to two Young tableaux

TableauToPoset

— Convert a Young tableau to a partially ordered set of coordinates

NumberOfTableaux

— Get a count of Young tableaux for a given size or shape

HookLengths

— Get the list of lists whose entries are the hook lengths of the entries of a Young tableau

StandardYoungTableaux

— Enumerate all standard Young tableaux of a given shape

RandomYoungTableau

— Generate a random Young tableau with a given shape

Multisets

CanonicalMultiset

— canonical form of a multiset

EnumerateMultisetPartialDerangements

— enumerate a multiset's partial derangements

Multichoose

— compute the multichoose operation

MultisetAssociation

— assign every element of a multiset a unique number, in effect turning a multiset into a set.

MultisetPartialDerangements

— list the partial derangements of a multiset one by one

Make a list of the elements one by one

OrderlessCombinations

— Generate all possible orderless lists of a given length with a given set of elements

OrderlessCombinationsOfUnmarkedElements

— Generate all possible orderless lists of a given length with a given set of unmarked indistinguishable plain unlabeled elements

Combinatorial Optimization and Existence

SelectPermutations

— Get permutations that satisfy a certain criterion

SelectSubsets

— Generate subsets that satisfy a certain criterion

SelectTuples

— Generate tuples that satisfy a certain criterion

Combinatorial Functions

CentralBinomialCoefficient

— compute the central binomial coefficient

EulerianCatalanNumber

— compute the Eulerian Catalan number

EulerianNumber

— Get the number of permutations with a given number of ascents

EulerianNumberOfTheSecondKind

— compute the Eulerian number of the second kind

NarayanaNumber

— compute the Narayana number

SecantNumber

— compute the secant number

, also referred to a zig number

SignedLahNumber

— compute the signed Lah number

L(n,k)

TangentNumber

— compute the tangent numbers, also referred to as zag numbers

UnsignedLahNumber

— compute the unsigned Lah number

L(n,k)

Encoding, Decoding, and Representation related

Fibbinary

— Get the fibbinary sequence

FibonacciEncode

— Fibonacci-digit encoding for a number

GrayCode

— Find the Gray code for an integer

HuffmanCodeWords

— Find optimal Huffman code words given a list of probabilities

HuffmanDecode

— Decode data specified by a Huffman encoding

HuffmanEncode

— Find a Huffman encoding for a string

InverseFibonacci

— Compute the inverse Fibonacci function

InverseGrayCode

— Find the integer corresponding to a given Gray code

ZeckendorfRepresentation

— Give the 0–1 list that indicates the unique nonconsecutive Fibonacci numbers that sum to the non-negative integer input

Permutation Functions

LehmerCodeFromPermutation

— Create the Lehmer code corresponding to a given permutation

NextPermutation

— Get the next permutation in lexicographic order

PermutationAscents

— Give the indices of a permutation where there is an immediate ascent

PermutationGraph

— Compute the permutation graph of a permutation

PermutationIndex

— Give the lexicographic index of a permutation

PermutationFromIndex

— Give the permutation corresponding to a given length and lexicographic index

PermutationMajorIndex

— Compute the major index of a permutation

PermutationToTableaux

— Get the two Young tableaux corresponding to a permutation

TableauxToPermutation

— Get a permutation corresponding to two Young tableaux

Permutation Inversion functions

FromInversionVector

— Construct the permutation list corresponding to the given inversion vector

InversionCount

— Count the number of pairs of out-of-order elements in a permutation

InversionVectorQ

— Check if a list is the inversion vector of a permutation written as a list

PermutationCountByInversions

— Get the number of permutations having a specified length and number of inversions

Permutation Runs and Ascents

DescendingSublists

— Split a list at its left-to-right maxima

PermutationAscents

— Give the indices of a permutation where there is an immediate ascent

EulerianNumber

— Get the number of permutations with a given number of ascents

EulerianNumberOfTheSecondKind

— compute the Eulerian number of the second kind

Order Theory

DivisorHasseDiagram

— Generate a transitive reduction graph for the divisors of a positive integer

HasseDiagram

— Construct a Hasse diagram of a poset

PartialOrderGraphQ

— Test whether a graph is a partial order, that is, reflexive, antisymmetric and transitive

PosetQ

— Determine if a set of coordinates is partially ordered

TableauToPoset

— Convert a Young tableau to a partially ordered set of coordinates

ReflexiveGraphQ

— Test whether a graph is reflexive

TransitiveGraphQ

— Test whether the binary relation defined by edges of a graph is transitive

Factorial Functions

GaussFactorial

— compute the Gauss factorial

Phitorial

— compute the phitorial

Primorial

— compute the primorial

Recurrence Functions

LucasNumberU1

— compute the term of the Lucas sequence of the first kind

LucasNumberV2

— compute the term of the Lucas sequence of the second kind

Q Analogues

QExponential

— q-exponential

QMultinomial

— q-multinomial

Modular Number Theory

RationalNumberRepeatingDecimalPeriod

— find the period of a repeating decimal of a rational number

Alternating Permutations

SecantNumber

— compute the secant number

, also referred to a zig number

TangentNumber

— compute the tangent numbers, also referred to as zag numbers

Examples

Basic Examples (4)

Find the millionth permutation with 10 items:

In[1]:=

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Out[1]=

To solve the challenge Lexicographic permutations at Project Euler, subtract 1 and join the digits with FromDigits:

In[2]:=

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Out[2]=

The googolth 111-tuple:

In[3]:=

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Out[3]=

Find some binomial representations for the number 403:

In[4]:=

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Out[4]=

Publisher

Peter Burbery

Disclosures

Set to load on startup with "Loading" -> "Startup".
Learn More »

Compatibility

Wolfram Language Version 13.3

Version History

License Information

BSD 3-Clause "New" or "Revised" License

Paclet Source

Source Metadata

Source: https://github.com/PeterCullenBurbery/combinatorics-paclet