| Type: | Package |
| Title: | Young Tableaux |
| Version: | 0.5.0 |
| Date: | 2024-07-22 |
| Author: | Stéphane Laurent |
| Maintainer: | Stéphane Laurent <laurent_step@outlook.fr> |
| Description: | Deals with Young tableaux (field of combinatorics). For standard Young tabeaux, performs enumeration, counting, random generation, the Robinson-Schensted correspondence, and conversion to and from paths on the Young lattice. Also performs enumeration and counting of semistandard Young tableaux, enumeration of skew semistandard Young tableaux, enumeration of Gelfand-Tsetlin patterns, and computation of Kostka numbers. |
| License: | GPL-3 |
| URL: | https://github.com/stla/syt |
| BugReports: | https://github.com/stla/syt/issues |
| Imports: | Matrix, partitions, utils |
| Suggests: | testthat |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.3.1 |
| NeedsCompilation: | no |
| Packaged: | 2024-07-23 09:05:45 UTC; User |
| Repository: | CRAN |
| Date/Publication: | 2024-07-23 20:10:01 UTC |
Gelfand-Tsetlin patterns
Description
Enumeration of Gelfand-Tsetlin patterns defined by a given integer partition and a given weight.
Usage
GelfandTsetlinPatterns(lambda, weight)
Arguments
lambda |
integer partition; up to trailing zeros, this will be the top diagonal of the generated Gelfand-Tsetlin patterns |
weight |
integer vector; the partial sums of this vector will be the diagonal sums of the generated Gelfand-Tsetlin patterns |
Value
A list of Gelfand-Tsetlin patterns. A Gelfand-Tsetlin pattern is a
triangular array of non-negative integers, and it is represented by the
list of its rows. Hence the first element of this list is an integer, the
second element is an integer vector of length two, and so on. The length
of this list is the length of weight.
See Also
Examples
GTpatterns <- GelfandTsetlinPatterns(c(3, 1), c(1, 1, 1, 1))
lapply(GTpatterns, prettyGT)
Kostka number
Description
Computes a Kostka number.
Usage
KostkaNumber(lambda, mu)
Arguments
lambda |
an integer partition |
mu |
an integer vector whose sum equals the weight (i.e. the sum) of
|
Details
The Kostka number K(\lambda,\mu) is the number of semistandard
Young tableaux with shape \lambda and weight \mu. It does
not depend on the order of the elements of \mu (so one can always
take an integer partition for \mu). The weight is the
vector whose i-th element is the number of occurrences of i
in the tableau.
Value
The Kostka number corresponding to lambda and mu.
See Also
KostkaNumbers, KostkaNumbersWithGivenMu,
KostkaNumbersWithGivenLambda,
skewKostkaNumbers.
Examples
KostkaNumber(c(3,2), c(1,1,1,2))
KostkaNumber(c(3,2), c(1,1,2,1))
KostkaNumber(c(3,2), c(1,2,1,1))
KostkaNumber(c(3,2), c(2,1,1,1))
lambda <- c(4, 3, 1)
mu <- rep(1, sum(lambda))
KostkaNumber(lambda, mu) == count_sytx(lambda) # should be TRUE
Kostka numbers for all partitions of a given weight
Description
Computes the Kostka numbers for all integer partitions of a given weight.
Usage
KostkaNumbers(n)
Arguments
n |
positive integer, the weight of the partitions |
Value
An integer matrix, whose row names and column names encode the
partitions \lambda and \mu and whose entries are the
Kostka numbers K(\lambda,\mu).
See Also
KostkaNumbersWithGivenLambda,
KostkaNumbersWithGivenMu, skewKostkaNumbers.
Examples
KostkaNumbers(4)
Kostka numbers with given \lambda
Description
Lists all positive Kostka numbers K(\lambda,\mu) with
a given partition \lambda.
Usage
KostkaNumbersWithGivenLambda(lambda, output = "vector")
Arguments
lambda |
integer partition |
output |
the format of the output, either |
Value
If output="vector", this function returns a named vector.
This vector is made of the non-zero (i.e. positive) Kostka numbers
K(\lambda,\mu), which are integers, and its names encode the
partitions \mu.
If output="list", this function returns a list of lists.
Each of these lists has two
elements. The first one is named mu and is an integer
partition, and the second one is named value and is a positive
integer, the Kostka number K(\lambda,\mu). It is faster to
compute the Kostka numbers with this function than computing the
individual Kostka numbers with the function KostkaNumber.
See Also
KostkaNumber, KostkaNumbers,
KostkaNumbersWithGivenMu.
Examples
KostkaNumbersWithGivenLambda(c(2, 1, 1))
Kostka numbers with given \mu
Description
Lists all positive Kostka numbers K(\lambda,\mu) with
a given partition \mu.
Usage
KostkaNumbersWithGivenMu(mu, output = "vector")
Arguments
mu |
integer partition |
output |
the format of the output, either |
Value
If output="vector", this function returns a named vector.
This vector is made of the positive Kostka numbers
K(\lambda,\mu) and its names encode the partitions \lambda.
If output="list", this function returns a list of lists.
Each of these lists has two
elements. The first one is named lambda and is an integer
partition, and the second one is named value and is a positive
integer, the Kostka number K(\lambda,\mu). It is faster to
compute the Kostka numbers with this function than computing the
individual Kostka numbers with the function KostkaNumber.
See Also
KostkaNumber, KostkaNumbers,
KostkaNumbersWithGivenLambda.
Examples
KostkaNumbersWithGivenMu(c(2, 1, 1))
Robinson-Schensted correspondence
Description
Pair of standard Young tableaux given from a permutation by the Robinson-Schensted correspondence.
Usage
RS(sigma)
Arguments
sigma |
a permutation given as a vector of integers |
Value
A list of two standard Young tableaux.
Examples
RS(c(1, 3, 6, 4, 7, 5, 2))
Semistandard skew tableaux
Description
Enumeration of all semistandard skew tableaux with given shape and given maximum entry.
Usage
all_ssSkewTableaux(lambda, mu, n)
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
n |
a positive integer, the maximum entry of the skew tableaux |
Value
The list of all semistandard skew tableaux whose shape is the skew
partition defined by lambda and mu and with maximum entry
n.
See Also
all_ssytx,
skewTableauxWithGivenShapeAndWeight.
Examples
ssstx <- all_ssSkewTableaux(c(4, 3, 1), c(2, 2), 2)
lapply(ssstx, prettySkewTableau)
Enumeration of semistandard Young tableaux
Description
Generates all semistandard Young tableaux of a given shape and
filled with integers between 1 and a given n.
Usage
all_ssytx(lambda, n)
Arguments
lambda |
an integer partition, the shape |
n |
an integer, the maximum value of the entries (the minimum value
is |
Value
List of all semistandard Young tableaux with shape lambda
and filled with integers between 1 and n.
See Also
ssytx_withGivenShapeAndWeight.
Examples
ssytx <- all_ssytx(c(2, 1), 3)
lapply(ssytx, prettyTableau)
Enumeration of standard Young tableaux
Description
Generates all standard Young tableaux of a given shape.
Usage
all_sytx(lambda)
Arguments
lambda |
the shape, an integer partition |
Value
A list of standard Young tableaux.
Examples
sytx <- all_sytx(c(5, 2))
lapply(sytx, prettyTableau)
Tableau as ballot sequence
Description
Converts a ballot sequence to its corresponding standard Young tableau.
Usage
ballot2syt(a)
Arguments
a |
ballot sequence |
Value
A standard Young tableau.
See Also
Examples
a <- c(1,1,2,3,2,1)
ballot2syt(a)
Number of semistandard Young tableaux
Description
Number of semistandard Young tableaux of a given shape and
filled with integers between 1 and a given n.
Usage
count_ssytx(lambda, n)
Arguments
lambda |
an integer partition, the shape |
n |
an integer, the maximum value of the entries (the minimum value
is |
Value
The number of semistandard Young tableaux with shape lambda
and filled with integers between 1 and n.
See Also
Examples
count_ssytx(c(4, 3, 3, 2), 5)
Number of standard Young tableaux
Description
Number of standard Young tableaux of a given shape.
Usage
count_sytx(lambda)
Arguments
lambda |
an integer partition, the shape |
Value
An integer, the number of standard Young tableaux of shape lambda.
See Also
Examples
count_sytx(c(5, 4, 1))
length(all_sytx(c(5, 4, 1)))
Dual skew tableau
Description
Returns the dual (skew) tableau of a skew tableau.
Usage
dualSkewTableau(skewTableau)
Arguments
skewTableau |
a skew tableau |
Value
A skew tableau.
Examples
tbl <- list(c(NA, NA, 1, 1), c(NA, 1), c(1, 2))
dtbl <- dualSkewTableau(tbl)
prettySkewTableau(dtbl)
Dual tableau
Description
The dual tableau of a tableau (mirror image to the main diagonal).
Usage
dualTableau(tableau)
Arguments
tableau |
a tableau |
Value
A tableau.
Examples
tbl <- list(c("a", "s", "e", "f"), c("f", "o"), c("u"))
dualTableau(tbl)
Dual tableau
Description
The dual standard Young tableau of a standard Young tableau.
Usage
dualsyt(syt)
Arguments
syt |
standard Young tableau |
Value
A standard Young tableau.
Examples
syt <- list(c(1,2,6), c(3,5), 4)
dualsyt(syt)
First tableau of a given shape
Description
Returns the "first" standard Young tableau of a given shape.
Usage
firstsyt(lambda)
Arguments
lambda |
the shape, an integer partition |
Value
A standard Young tableau.
Examples
firstsyt(c(4, 2, 1))
Growth process to tableau
Description
Converts a growth process of integer partitions to its corresponding standard Young tableau.
Usage
gprocess2syt(path)
Arguments
path |
a path of the Young graph from the root vertex, given as a list of integer partitions |
Value
A standard Young tableau.
See Also
Examples
path <- list(1, 2, c(2,1), c(3,1), c(3,1,1))
gprocess2syt(path)
Hook lengths
Description
Hook lengths of a given integer partition.
Usage
hooklengths(lambda)
Arguments
lambda |
an integer partition |
Value
The hook lengths of the partition, given in a list.
See Also
Examples
hooklengths(c(4, 2))
Hooks
Description
Hooks of a given integer partition.
Usage
hooks(lambda)
Arguments
lambda |
integer partition |
Value
The hooks of the partition in a list.
See Also
Examples
hooks(c(4, 2))
Checks whether a tableau is semistandard
Description
Checks whether a tableau is a semistandard Young tableau.
Usage
isSSYT(tableau)
Arguments
tableau |
a tableau |
Value
A Boolean value.
Examples
tbl <- list(c(1, 2, 6), c(5, 5), 7)
isSSYT(tbl)
Checks whether a tableau is standard
Description
Checks whether a tableau is a standard Young tableau.
Usage
isSYT(tableau)
Arguments
tableau |
a tableau |
Value
A Boolean value.
Examples
tbl <- list(c(1, 2, 6), c(3, 5), 4)
isSYT(tbl)
Check whether a skew tableau is semistandard
Description
Check whether a skew tableau is a semistandard skew tableau.
Usage
isSemistandardSkewTableau(skewTableau)
Arguments
skewTableau |
a skew tableau |
Value
A Boolean value.
Examples
tbl <- list(c(NA, NA, 1, 1), c(NA, 1), c(1, 2))
isSemistandardSkewTableau(tbl)
Check whether a tableau is a skew tableau
Description
Check whether a tableau is a skew tableau.
Usage
isSkewTableau(tableau)
Arguments
tableau |
a tableau |
Value
A Boolean value.
Examples
tbl <- list(c(NA, NA, 1, 1), c(NA, 1), c(1, 2))
isSkewTableau(tbl)
Check whether a skew tableau is standard
Description
Check whether a skew tableau is a standard skew tableau.
Usage
isStandardSkewTableau(skewTableau)
Arguments
skewTableau |
a skew tableau |
Value
A Boolean value.
Examples
tbl <- list(c(NA, NA, 1, 1), c(NA, 1), c(1, 2))
isStandardSkewTableau(tbl)
Standard Young tableau from a matrix
Description
Converts a matrix to a standard Young tableau.
Usage
matrix2syt(M)
Arguments
M |
a matrix |
Value
A standard Young tableau.
See Also
Examples
M <- rbind(c(1,2,6), c(3,5,0), c(4,0,0))
matrix2syt(M)
Next tableau
Description
Given a standard Young tableau, returns the "next" one having the same shape.
Usage
nextsyt(syt)
Arguments
syt |
a standard Young tableau |
Value
A standard Young tableau of the same shape as syt, or
NULL if syt is the last standard Young tableau of this shape.
Examples
syt <- firstsyt(c(4, 2, 1))
nextsyt(syt)
Pretty Gelfand-Tsetlin pattern
Description
Pretty form of a Gelfand-Tsetlin pattern.
Usage
prettyGT(GT)
Arguments
GT |
a Gelfand-Tsetlin pattern |
Value
A 'noquote' character matrix.
Pretty skew tableau
Description
Pretty form of a skew tableau.
Usage
prettySkewTableau(skewTableau)
Arguments
skewTableau |
a skew tableau |
Value
A 'noquote' character matrix.
Examples
tbl <- list(c(NA, NA, 1, 1), c(NA, 1), c(1, 2))
prettySkewTableau(tbl)
Pretty tableau
Description
Pretty form of a tableau.
Usage
prettyTableau(tableau)
Arguments
tableau |
a tableau |
Value
A 'noquote' character matrix.
Examples
tbl <- list(c(0, 2, 1, 1), c(4, 1), c(1, 2))
prettyTableau(tbl)
Plancherel growth process
Description
Samples a path of the Young graph according to the Plancherel growth process.
Usage
rgprocess(n)
Arguments
n |
the size of the path to be sampled |
Value
The path as a list, starting from the root vertex 1.
See Also
gprocess2syt and syt2gprocess to convert
a Young path to a standard Young tableau and conversely.
Examples
rgprocess(7)
Random standard Young tableau
Description
Uniform sampling of a standard Young tableau of a given shape.
Usage
rsyt(lambda)
Arguments
lambda |
shape, an integer partition |
Value
A standard Young tableau of shape lambda.
Examples
rsyt(c(7, 3, 1))
Skew Gelfand-Tsetlin patterns
Description
Enumeration of Gelfand-Tsetlin patterns defined by a given skew partition and a given weight.
Usage
skewGelfandTsetlinPatterns(lambda, mu, weight)
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
weight |
integer vector; this vector will be the
differences of the row sums of the generated Gelfand-Tsetlin patterns;
consequently, there will be no generated Gelfand-Tsetlin pattern unless
the sum of |
Value
A list of matrices with non-negative integer entries. The number
of columns of these matrices is the length of lambda and the
number of rows of these matrices is one plus the length of weight.
See Also
Examples
skewGelfandTsetlinPatterns(c(3, 1, 1), c(2), c(1, 1, 1))
Skew Kostka numbers
Description
Skew Kostka numbers associated to a given skew partition.
Usage
skewKostkaNumbers(lambda, mu, output = "vector")
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
output |
the format of the output, either |
Details
The skew Kostka number K_{\lambda/\mu,\nu} is the number of skew
semistandard Young tableaux with shape \lambda/\mu and weight
\nu. The weight of a Young tableau is the
vector whose i-th element is the number of occurrences of i
in this tableau.
Value
If output="vector", the function returns a named vector.
This vector is made of the positive skew Kostka numbers
K_{\lambda/\mu,\nu} and its names encode the partitions \nu.
If ouput="list", the function returns a list. Each element of this
list is a named list with two elements: an integer partition \nu
in the field named "nu", and the corresponding skew Kostka number
K_{\lambda/\mu,\nu} in the field named "value". Only the
non-null skew Kostka numbers are provided by this list.
See Also
KostkaNumber, KostkaNumbersWithGivenMu.
Examples
skewKostkaNumbers(c(4,2,2), c(2,2))
Skew semistandard tableaux with given shape and weight
Description
Enumeration of all skew semistandard tableaux with a given
shape and a given weight. The weight of a tableau is the
vector whose i-th element is the number of occurrences of i
in this tableau.
Usage
skewTableauxWithGivenShapeAndWeight(lambda, mu, weight)
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
weight |
integer vector, the weight |
Value
List of all skew semistandard tableaux whose shape is the skew
partition defined by lambda and mu and whose weight is
weight.
Examples
ssstx <- skewTableauxWithGivenShapeAndWeight(c(3, 1, 1), c(2), c(1, 1, 1))
lapply(ssstx, prettySkewTableau)
Semistandard Young tableaux with given shape and weight
Description
Enumeration of all semistandard Young tableaux with a given
shape and a given weight. The weight of a tableau is the
vector whose i-th element is the number of occurrences of i
in this tableau.
Usage
ssytx_withGivenShapeAndWeight(lambda, weight)
Arguments
lambda |
integer partition, the shape |
weight |
integer vector, the weight |
Value
List of all semistandard Young tableaux with shape lambda
and weight weight.
See Also
Examples
ssytx <- ssytx_withGivenShapeAndWeight(c(4, 1), c(0, 2, 1, 1, 1))
lapply(ssytx, prettyTableau)
Tableau as ballot sequence
Description
Converts a standard Young tableau to its corresponding ballot sequence.
Usage
syt2ballot(syt)
Arguments
syt |
standard Young tableau |
Value
A ballot sequence.
See Also
Examples
syt <- list(c(1,2,6), c(3,5), 4)
syt2ballot(syt)
Tableau as growth process
Description
Converts a standard Young tableau to its corresponding growth process of partitions.
Usage
syt2gprocess(syt)
Arguments
syt |
standard Young tableau |
Value
A list of integer partitions, representing a path of the Young graph starting from the root vertex.
See Also
Examples
syt <- list(c(1,2,4), 3, 5)
syt2gprocess(syt)
Standard Young tableau as sparse matrix
Description
Representation of a standard Young tableau as a sparse matrix.
Usage
syt2matrix(syt)
Arguments
syt |
a standard Young tableau |
Value
A sparse matrix.
Note
This function is the same as tableau2matrix except that
in addition it checks that the given tableau is a standard Young tableau.
See Also
Examples
syt <- list(c(1, 2, 6), c(3, 5), 4)
syt2matrix(syt)
Tableau as sparse matrix
Description
Representation of a tableau as a sparse matrix; only for a tableau with numeric or logical entries.
Usage
tableau2matrix(tableau)
Arguments
tableau |
a tableau with numeric or logical entries |
Value
A sparse matrix.
Examples
syt <- list(c(1, 2, 6), c(3, 5), 4)
tableau2matrix(syt)
Shape of a tableau
Description
The shape of a tableau.
Usage
tableauShape(tableau)
Arguments
tableau |
a tableau (list of vectors having the same mode) |
Value
The shape of the tableau. This is an integer partition whose
i-th part is the number of boxes in the i-th row of
the tableau.
Examples
tableau <- list(c(2, 1, 3), c(5, 2))
tableauShape(tableau)