StandardYoungTableaux

Overview

Some stuff for standard Young tableaux:

generation of all SYTx of a given shape
count of all SYTx of a given shape
uniform sampling of SYTx
Robinson-Schensted(-Knuth) correspondence
conversion to and from ballot sequences
conversion to and from paths of integer partitions on the Young graph
Plancherel growth process

julia> syt = StandardYoungTableau([[1,2], [3,4], [5]])
julia> SYT2YoungPath(syt)
5-element Array{IPartition,1}:
 IPartition(1, [1])
 IPartition(2, [2])
 IPartition(3, [2, 1])
 IPartition(4, [2, 2])
 IPartition(5, [2, 2, 1])

Member functions

StandardYoungTableaux.IPartition — Type

IPartition(partition)

Defines an integer partition.

Argument

partition: vector of positive integers in decreasing order

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StandardYoungTableaux.StandardYoungTableau — Type

StandardYoungTableau(tableau)

Defines a standard Young tableau.

Argument

tableau: a vector of vectors of positive integers

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StandardYoungTableaux.RS — Method

RS(sigma)

Pair of standard Young tableaux given by the Robinson-Schensted(-Knuth) correspondence from a permutation.

Argument

sigma: a permutation given as a vector of integers

Example

RS([3, 4, 1, 2])

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StandardYoungTableaux.SYT2YoungPath — Method

SYT2YoungPath(syt)

Path of the Young graph corresponding to a standard Young tableau.

Argument

syt: a standard Young tableau

Example

y = StandardYoungTableau([[1,3,4], [2]])
SYT2YoungPath(y)

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StandardYoungTableaux.SYT2ballot — Method

SYT2ballot(syt)

The ballot sequence corresponding to a standard Young tableau.

Argument

syt: a standard Young tableau

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StandardYoungTableaux.YoungPath2SYT — Method

YoungPath2SYT(path)

Standard Young tableau corresponding to a path of the Young graph.

Argument

path: a vector of integers partitions forming a path of the Young graph

Example

y = StandardYoungTableau([[1,3,4], [2]])
path = SYT2YoungPath(y)
YoungPath2SYT(path)

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StandardYoungTableaux.allBallotSequences — Method

allBallotSequences(lambda)

All ballot sequences associated to a given integer partition.

Argument

lambda: an integer partition

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StandardYoungTableaux.allSYTx — Method

allSYTx(lambda)

All standard Young tableaux of a given shape.

Argument

lambda: an integer partition

Example

lambda = IPartition([3, 1])
allSYTx(lambda)

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StandardYoungTableaux.ballot2SYT — Method

ballot2SYT(b)

The standard Young tableau corresponding to a ballot sequence.

Argument

b: a ballot sequence (vector of integers)

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StandardYoungTableaux.countSYTx — Method

countSYTx(lambda)

Number of standard Young tableaux of a given shape.

Argument

lambda: an integer partition, the shape

Example

lambda = IPartition([4, 2])
countSYTx(lambda)
length(allSYTx(lambda))

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StandardYoungTableaux.dualSYT — Method

dualSYT(syt)

Dual (conjugate) standard Young tableau.

Argument

syt: a standard Young tableau

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StandardYoungTableaux.firstSYT — Method

firstSYT(lambda)

The 'first' standard Young tableau of a given shape.

Argument

lambda: an integer partition

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StandardYoungTableaux.hooklengths — Method

hooklengths(lambda)

The hook lengths of a partition.

Argument

lambda: an integer partition

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StandardYoungTableaux.hooks — Method

hooks(lambda)

The hooks of a partition.

Argument

lambda: an integer partition

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StandardYoungTableaux.nextSYT — Method

nextSYT(syt)

The 'next' standard Young tableau.

Argument

syt: a standard Young tableau

Example

y = firstSYT(IPartition([4, 2]))
nextSYT(y)

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StandardYoungTableaux.randomSYT — Method

randomSYT(lambda)

Uniformly random standard Young tableau of a given shape.

Argument

lambda: an integer partition, the shape

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StandardYoungTableaux.randomYoungPath — Method

randomYoungPath(n)

Samples a path of the Young graph according to the Plancherel growth process.

Argument

n: a positive integer, the size of the path to be sampled

Example

path = randomYoungPath(5)
YoungPath2SYT(path)

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