h*x t      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`a1.4.5.0 Safe-Inferred*bcdefghijklmnopqrstuvwxyz{|}~Evaluation of Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-InferredjackpolynomialsEvaluation of Jack polynomialjackpolynomialsEvaluation of Jack polynomialjackpolynomialsEvaluation of zonal polynomialjackpolynomialsEvaluation of zonal polynomialjackpolynomialsEvaluation of Schur polynomialjackpolynomialsEvaluation of Schur polynomialjackpolynomials%Evaluation of a skew Schur polynomialjackpolynomials%Evaluation of a skew Schur polynomialjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsvalues of the variablesjackpolynomialspartition of integers jackpolynomialsvalues of the variablesjackpolynomialspartition of integers jackpolynomialsvalues of the variablesjackpolynomials)the outer partition of the skew partitionjackpolynomials)the inner partition of the skew partitionjackpolynomialsvalues of the variablesjackpolynomials)the outer partition of the skew partitionjackpolynomials)the inner partition of the skew partition   Safe-Inferred 7 jackpolynomialsInefficient hypergeometric function of a matrix argument (for testing purpose)  Symbolic Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-InferredR jackpolynomialsJack polynomial jackpolynomialsJack polynomial jackpolynomialsSkew Jack polynomial jackpolynomialsSkew Jack polynomialjackpolynomialsZonal polynomialjackpolynomialsZonal polynomialjackpolynomialsSkew zonal polynomialjackpolynomialsZonal polynomialjackpolynomialsSchur polynomialjackpolynomialsSchur polynomialjackpolynomialsSkew Schur polynomialjackpolynomialsSkew Schur polynomial jackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Q jackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Q jackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsJack parameterjackpolynomialswhich skew Jack polynomial, J, C, P or Q jackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsJack parameterjackpolynomialswhich skew Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partition    .Jack polynomials with symbolic Jack parameter.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferredjackpolynomials,Jack polynomial with symbolic Jack parameterjackpolynomials,Jack polynomial with symbolic Jack parameterjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialswhich skew Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialswhich skew Jack polynomial, J, C, P or Q$Some utilities for Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferredt?jackpolynomialsmonomial symmetric polynomialjackpolynomialsMonomial symmetric polynomial/putStrLn $ prettySpray' (msPolynomial 3 [2, 1])(1) x1^2.x2 + (1) x1^2.x3 + (1) x1.x2^2 + (1) x1.x3^2 + (1) x2^2.x3 + (1) x2.x3^2jackpolynomials6Checks whether a spray defines a symmetric polynomial.-- note that the sum of two symmetric polynomials is not symmetric/-- if they have different numbers of variables:1spray = schurPol' 4 [2, 2] ^+^ schurPol' 3 [2, 1]isSymmetricSpray sprayjackpolynomialsSymmetric polynomial as a linear combination of monomial symmetric polynomials.jackpolynomialsPrints a symmetric spray as a linear combination of monomial symmetric polynomials:putStrLn $ prettySymmetricNumSpray $ schurPol' 3 [3, 1, 1]M[3,1,1] + M[2,2,1]jackpolynomialsPrints a symmetric spray as a linear combination of monomial symmetric polynomials=putStrLn $ prettySymmetricQSpray $ jackPol' 3 [3, 1, 1] 2 'J'42*M[3,1,1] + 28*M[2,2,1]jackpolynomialsSame as  but for a  symmetric spray jackpolynomialsPrints a symmetric parametric spray as a linear combination of monomial symmetric polynomials.putStrLn $ prettySymmetricParametricQSpray ["a"] $ jackSymbolicPol' 3 [3, 1, 1] 'J'={ [ 4*a^2 + 10*a + 6 ] }*M[3,1,1] + { [ 8*a + 12 ] }*M[2,2,1]!jackpolynomialsPrints a symmetric simple parametric spray as a linear combination of monomial symmetric polynomials."jackpolynomialsLaplace-Beltrami operator on the space of homogeneous symmetric polynomials; neither symmetry and homogeneity are checked.#jackpolynomialsCalogero-Sutherland operator on the space of homogeneous symmetric polynomials; neither symmetry and homogeneity are checked$jackpolynomialsPower sum polynomial/putStrLn $ prettyQSpray (psPolynomial 3 [2, 1])?x^3 + x^2.y + x^2.z + x.y^2 + x.z^2 + y^3 + y^2.z + y.z^2 + z^3jackpolynomialsmonomial symmetric polynomial as a linear combination of power sum polynomialsjackpolynomials$the factor in the Hall inner productjackpolynomialssymmetric polynomial as a linear combination of power sum polynomials%jackpolynomialsSymmetric polynomial as a linear combination of power sum polynomials. Symmetry is not checked.&jackpolynomialsSymmetric polynomial as a linear combination of power sum polynomials. Same as  psCombination: but with other constraints on the base ring of the spray.'jackpolynomialsSymmetric parametric spray as a linear combination of power sum polynomials. jackpolynomials%the Hall inner product with parameter(jackpolynomialsHall inner product with Jack parameter, aka Jack-scalar product. It makes sense only for symmetric sprays, and the symmetry is not checked. )jackpolynomials+Hall inner product with parameter. Same as hallInnerProduct= but with other constraints on the base ring of the sprays.*jackpolynomials+Hall inner product with parameter. Same as hallInnerProduct but with other constraints on the base ring of the sprays. It is applicable to  Spray Int sprays.+jackpolynomialsHall inner product with parameter for parametric sprays, because the type of the parameter in hallInnerProduct is strange. For example, a ParametricQSpray spray is a Spray RatioOfQSprays spray, and it makes more sense to compute the Hall product with a Rational5 parameter then to compute the Hall product with a RatioOfQSprays parameter.-import Math.Algebra.Jack.SymmetricPolynomials#import Math.Algebra.JackSymbolicPolimport Math.Algebra.Hspray!jp = jackSymbolicPol 3 [2, 1] 'P'hallInnerProduct''' jp jp 5 == hallInnerProduct jp jp (constantRatioOfSprays 5),jackpolynomialsHall inner product with parameter for parametric sprays. Same as hallInnerProduct''' but with other constraints on the types. It is applicable to SimpleParametricQSpray sprays, while hallInnerProduct''' is not.jackpolynomials.the Hall inner product with symbolic parameter-jackpolynomialsHall inner product with symbolic parameter. See README for some examples..jackpolynomials4Hall inner product with symbolic parameter. Same as symbolicHallInnerProduct# but with other type constraints./jackpolynomials4Hall inner product with symbolic parameter. Same as symbolicHallInnerProduct8 but with other type constraints. It is applicable to  Spray Int sprays.0jackpolynomials)Complete symmetric homogeneous polynomial0putStrLn $ prettyQSpray (cshPolynomial 3 [2, 1])x^3 + 2*x^2.y + 2*x^2.z + 2*x.y^2 + 3*x.y.z + 2*x.z^2 + y^3 + 2*y^2.z + 2*y.z^2 + z^3jackpolynomialspower sum polynomial as a linear combination of complete symmetric homogeneous polynomialsjackpolynomialsmonomial symmetric polynomial as a linear combination of complete symmetric homogeneous polynomialsjackpolynomialssymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials1jackpolynomialsSymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials. Symmetry is not checked.2jackpolynomialsSymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials. Same as cshCombination: but with other constraints on the base ring of the spray.3jackpolynomials Elementary symmetric polynomial./putStrLn $ prettyQSpray (esPolynomial 3 [2, 1])7x^2.y + x^2.z + x.y^2 + 3*x.y.z + x.z^2 + y^2.z + y.z^2jackpolynomialspower sum polynomial as a linear combination of elementary symmetric polynomialsjackpolynomialsmonomial symmetric polynomial as a linear combination of elementary symmetric polynomialsjackpolynomialssymmetric polynomial as a linear combination of elementary symmetric polynomials4jackpolynomialsSymmetric polynomial as a linear combination of elementary symmetric polynomials. Symmetry is not checked.5jackpolynomialsSymmetric polynomial as a linear combination of elementary symmetric polynomials. Same as  esCombination: but with other constraints on the base ring of the spray.jackpolynomialscomplete symmetric homogeneous polynomial as a linear combination of Schur polynomialsjackpolynomialssymmetric polynomial as a linear combination of Schur polynomials6jackpolynomialsSymmetric polynomial as a linear combination of Schur polynomials. Symmetry is not checked.7jackpolynomialsSymmetric polynomial as a linear combination of Schur polynomials. Same as schurCombination: but with other constraints on the base ring of the spray.8jackpolynomialsKostka numbers K_{\lambda,\mu}(\alpha)( for a given weight of the partitions \lambda and \mu and a given Jack parameter \alpha. (these are the standard Kostka numbers when \alpha=1<). This returns a map whose keys represent the partitions \lambda' and the value attached to a partition \lambda represents the map #\mu \mapsto K_{\lambda,\mu}(\alpha) where the partition \mu6 is included in the keys of this map if and only if K_{\lambda,\mu}(\alpha) \neq 0.9jackpolynomialsKostka numbers K_{\lambda,\mu}(\alpha) with symbolic Jack parameter \alpha( for a given weight of the partitions \lambda and \mu=. This returns a map whose keys represent the partitions \lambda' and the value attached to a partition \lambda represents the map #\mu \mapsto K_{\lambda,\mu}(\alpha) where the partition \mu6 is included in the keys of this map if and only if K_{\lambda,\mu}(\alpha) \neq 0.:jackpolynomialsSkew Kostka numbers K_{\lambda/\mu, \nu}(\alpha) with a given Jack parameter \alpha and a given skew partition  \lambda/\mu;. This returns a map whose keys represent the partitions \nu.;jackpolynomialsSkew Kostka numbers K_{\lambda/\mu, \nu}(\alpha) with symbolic Jack parameter \alpha for a given skew partition  \lambda/\mu;. This returns a map whose keys represent the partitions \nu.jackpolynomials8monomial symmetric polynomials in Jack polynomials basisjackpolynomials8monomial symmetric polynomials in Jack polynomials basis<jackpolynomialsSymmetric polynomial as a linear combination of Jack polynomials with a given Jack parameter. Symmetry is not checked.=jackpolynomialsSymmetric polynomial as a linear combination of Jack polynomials with symbolic parameter. Symmetry is not checked.>jackpolynomialsSymmetric parametric polynomial as a linear combination of Jack polynomials with symbolic parameter. Similar to jackSymbolicCombination but for a parametric spray.?jackpolynomialsKostka-Foulkes polynomial of two given partitions. This is a univariate polynomial whose value at 1, is the Kostka number of the two partitions.@jackpolynomialsKostka-Foulkes polynomial of two given partitions. This is a univariate polynomial whose value at 1, is the Kostka number of the two partitions.AjackpolynomialsSkew Kostka-Foulkes polynomial. This is a univariate polynomial associated to a skew partition and a partition, and its value at 1< is the skew Kostka number associated to these partitions.BjackpolynomialsSkew Kostka-Foulkes polynomial. This is a univariate polynomial associated to a skew partition and a partition, and its value at 1< is the skew Kostka number associated to these partitions.Cjackpolynomialsqt-Kostka polynomials, aka Kostka-Macdonald polynomials. They are usually denoted by K(\lambda, \mu) for two integer partitions \lambda and \mu. For a given partition \mu(, the function returns the polynomials K(\lambda, \mu) for all partitions \lambda of the same weight as \mu.Djackpolynomialsqt-Kostka polynomials, aka Kostka-Macdonald polynomials. They are usually denoted by K(\lambda, \mu) for two integer partitions \lambda and mu. For a given partition \mu(, the function returns the polynomials K(\lambda, \mu) for all partitions \lambda of the same weight as \mu.Ejackpolynomials9Skew qt-Kostka polynomials. They are usually denoted by K(\lambda/\mu, \nu) for two integer partitions \lambda and \mu4 defining a skew partition and an integer partition \nu. For given partitions \lambda and \mu(, the function returns the polynomials K(\lambda/\mu, \nu) for all partitions \nu, of the same weight as the skew partition.Fjackpolynomials9Skew qt-Kostka polynomials. They are usually denoted by K(\lambda/\mu, \nu) for two integer partitions \lambda and \mu4 defining a skew partition and an integer partition \nu. For given partitions \lambda and \mu(, the function returns the polynomials K(\lambda/\mu, \nu) for all partitions \nu, of the same weight as the skew partition.GjackpolynomialsHall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter.HjackpolynomialsHall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter.IjackpolynomialsHall-Littlewood polynomials as linear combinations of Schur polynomials.JjackpolynomialsSkew Hall-Littlewood polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter.KjackpolynomialsSkew Hall-Littlewood polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.Ljackpolynomialst-Schur polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter.Mjackpolynomialst-Schur polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.NjackpolynomialsSkew t-Schur polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.OjackpolynomialsSkew t-Schur polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.PjackpolynomialsMacdonald polynomial. This is a symmetric multivariate polynomial depending on two parameters usually denoted by q and t.*macPoly = macdonaldPolynomial 3 [2, 1] 'P'=putStrLn $ prettySymmetricParametricQSpray ["q", "t"] macPoly{ [ 1 ] }*M[2,1] + { [ 2*q.t^2 - q.t - q + t^2 + t - 2 ] %//% [ q.t^2 - 1 ] }*M[1,1,1]QjackpolynomialsMacdonald polynomial. This is a symmetric multivariate polynomial depending on two parameters usually denoted by q and t.RjackpolynomialsSkew Macdonald polynomial of a given skew partition. This is a multivariate symmetric polynomial with two parameters.SjackpolynomialsSkew Macdonald polynomial of a given skew partition. This is a multivariate symmetric polynomial with two parameters.TjackpolynomialsMacdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in two parameters.UjackpolynomialsMacdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in two parameters.VjackpolynomialsSkew Macdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients depend on two parameters.WjackpolynomialsSkew Macdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients depend on two parameters.XjackpolynomialsModified Macdonald polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomials in two parameters.YjackpolynomialsModified Macdonald polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomials in two parameters.ZjackpolynomialsFlagged Schur polynomial. A flagged Schur polynomial is not symmetric in general.[jackpolynomialsFlagged Schur polynomial. A flagged Schur polynomial is not symmetric in general.\jackpolynomialsFlagged skew Schur polynomial. A flagged skew Schur polynomial is not symmetric in general.]jackpolynomialsFlagged skew Schur polynomial. A flagged skew Schur polynomial is not symmetric in general.^jackpolynomials!Factorial Schur polynomial. See  https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r84/pdfKreiman's paper %Products of factorial Schur functions for the definition._jackpolynomials!Factorial Schur polynomial. See  https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r84/pdfKreiman's paper %Products of factorial Schur functions for the definition.`jackpolynomials'Skew factorial Schur polynomial. See  https://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/opapers/s28macdonald.pdfMacdonald's paper %Schur functions: theme and variations$, 6th variation, for the definition.ajackpolynomials'Skew factorial Schur polynomial. See  https://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/opapers/s28macdonald.pdfMacdonald's paper %Schur functions: theme and variations$, 6th variation, for the definition.1jackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsnumber of variablesjackpolynomialsinteger partition$jackpolynomialsnumber of variablesjackpolynomialsinteger partition'jackpolynomialsparametric sprayjackpolynomialssprayjackpolynomialssprayjackpolynomials parameter(jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter)jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter*jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter+jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter,jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter0jackpolynomialsnumber of variablesjackpolynomialsinteger partition3jackpolynomialsnumber of variablesjackpolynomialsinteger partition8jackpolynomialsweight of the partitionsjackpolynomialsJack parameter:jackpolynomialsJack parameterjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partition;jackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partition<jackpolynomialsJack parameterjackpolynomialswhich Jack polynomials, J, C, P or Qjackpolynomials)spray representing a symmetric polynomialjackpolynomials5map representing the linear combination; a partition lambda1 in the keys of this map corresponds to the term &coeff *^ jackPol' n lambda alpha which, where coeff' is the value attached to this key and n( is the number of variables of the spray=jackpolynomialswhich Jack polynomials, J, C, P or Qjackpolynomials)spray representing a symmetric polynomialjackpolynomials5map representing the linear combination; a partition lambda1 in the keys of this map corresponds to the term (coeff *^ jackSymbolicPol' n lambda which, where coeff' is the value attached to this key and n( is the number of variables of the spray>jackpolynomialswhich Jack polynomials, J, C, P or Qjackpolynomials4parametric spray representing a symmetric polynomialjackpolynomials5map representing the linear combination; a partition lambda1 in the keys of this map corresponds to the term (coeff *^ jackSymbolicPol' n lambda which, where coeff' is the value attached to this key and n( is the number of variables of the sprayAjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsinteger partition; the equality of the weight of this partition with the weight of the skew partition is a necessary condition to get a non-zero polynomialBjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsinteger partition; the equality of the weight of this partition with the weight of the skew partition is a necessary condition to get a non-zero polynomialEjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionFjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionGjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomials"which Hall-Littlewood polynomial, P or QHjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomials"which Hall-Littlewood polynomial, P or QIjackpolynomials;weight of the partitions of the Hall-Littlewood polynomialsjackpolynomials#which Hall-Littlewood polynomials, P or QJjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials'which skew Hall-Littlewood polynomial, P or QKjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials'which skew Hall-Littlewood polynomial, P or QLjackpolynomialsnumber of variablesjackpolynomialsinteger partitionMjackpolynomialsnumber of variablesjackpolynomialsinteger partitionNjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionOjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionPjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialswhich Macdonald polynomial, P or QQjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialswhich Macdonald polynomial, P or QRjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials!which skew Macdonald polynomial, P or QSjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials!which skew Macdonald polynomial, P or QTjackpolynomialsnumber of variablesjackpolynomialsinteger partitionUjackpolynomialsnumber of variablesjackpolynomialsinteger partitionVjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionWjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionXjackpolynomialsnumber of variablesjackpolynomialsinteger partitionYjackpolynomialsnumber of variablesjackpolynomialsinteger partitionZjackpolynomialsinteger partitionjackpolynomials lower boundsjackpolynomials upper bounds[jackpolynomialsinteger partitionjackpolynomials lower boundsjackpolynomials upper bounds\jackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials lower boundsjackpolynomials upper bounds]jackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials lower boundsjackpolynomials upper bounds^jackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsthe sequence denoted by y in the reference paper _jackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsthe sequence denoted by y in the reference paper`jackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsthe sequence denoted by a in the reference paperajackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsthe sequence denoted by a in the reference paper$03%&'124567<=> !"#()*+,-./89:;?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`a$03%&'124567<=> !"#()*+,-./89:;?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`a      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~.jackpolynomials-1.4.5.0-98RUkLlW11RFOedD9EEdUfMath.Algebra.JackMath.Algebra.Jack.HypergeoPQMath.Algebra.JackPolMath.Algebra.JackSymbolicPol!Math.Algebra.SymmetricPolynomialsjackpolynomialsMath.Algebra.Jack.Internal Partitionjack'jackzonal'zonalschur'schur skewSchur' skewSchur hypergeoPQjackPol'jackPol skewJackPol skewJackPol' zonalPol'zonalPol skewZonalPol' skewZonalPol schurPol'schurPol skewSchurPol' skewSchurPoljackSymbolicPol'jackSymbolicPolskewJackSymbolicPolskewJackSymbolicPol' msPolynomialisSymmetricSpray msCombinationprettySymmetricNumSprayprettySymmetricQSprayprettySymmetricQSpray'prettySymmetricParametricQSpray%prettySymmetricSimpleParametricQSpraylaplaceBeltramicalogeroSutherland psPolynomial psCombinationpsCombination'psCombination''hallInnerProducthallInnerProduct'hallInnerProduct''hallInnerProduct'''hallInnerProduct''''symbolicHallInnerProductsymbolicHallInnerProduct'symbolicHallInnerProduct'' cshPolynomialcshCombinationcshCombination' esPolynomial esCombinationesCombination'schurCombinationschurCombination' kostkaNumberssymbolicKostkaNumbersskewKostkaNumberssymbolicSkewKostkaNumbersjackCombinationjackSymbolicCombinationjackSymbolicCombination'kostkaFoulkesPolynomialkostkaFoulkesPolynomial'skewKostkaFoulkesPolynomialskewKostkaFoulkesPolynomial'qtKostkaPolynomialsqtKostkaPolynomials'qtSkewKostkaPolynomialsqtSkewKostkaPolynomials'hallLittlewoodPolynomialhallLittlewoodPolynomial' transitionsSchurToHallLittlewoodskewHallLittlewoodPolynomialskewHallLittlewoodPolynomial'tSchurPolynomialtSchurPolynomial'tSkewSchurPolynomialtSkewSchurPolynomial'macdonaldPolynomialmacdonaldPolynomial'skewMacdonaldPolynomialskewMacdonaldPolynomial'macdonaldJpolynomialmacdonaldJpolynomial'skewMacdonaldJpolynomialskewMacdonaldJpolynomial'modifiedMacdonaldPolynomialmodifiedMacdonaldPolynomial'flaggedSchurPolflaggedSchurPol'flaggedSkewSchurPolflaggedSkewSchurPol'factorialSchurPolfactorialSchurPol'skewFactorialSchurPolskewFactorialSchurPol' jackCoeffP jackCoeffQ jackCoeffCjackSymbolicCoeffCjackSymbolicCoeffPinvjackSymbolicCoeffQinv _betaratio_betaRatioOfSprays _isPartition_N_fromIntskewSchurLRCoefficientsisSkewPartition sprayToMap comboToSpray_kostkaNumbers_inverseKostkaMatrix_symbolicKostkaNumbers_inverseSymbolicKostkaMatrix_kostkaFoulkesPolynomial&_hallLittlewoodPolynomialsInSchurBasis$_transitionMatrixHallLittlewoodSchurskewHallLittlewoodPskewHallLittlewoodQ flaggedSemiStandardYoungTableaux tableauWeight isIncreasingflaggedSkewTableauxskewTableauWeight_skewKostkaFoulkesPolynomialmacdonaldPolynomialPmacdonaldPolynomialQskewMacdonaldPolynomialPskewMacdonaldPolynomialQchi_lambda_mu_rhoclambda clambdamumacdonaldJinMSPbasisinverseKostkaNumbersskewSymbolicJackInMSPbasisskewJackInMSPbasismsPolynomialUnsafe%hspray-0.5.4.0-L4Ha9KF0p3RGToxE8j6HqkMath.Algebra.HsprayQSpray' mspInPSbasiszlambda_psCombination_hallInnerProduct_symbolicHallInnerProduct pspInCSHbasis mspInCSHbasis_cshCombination pspInESbasis mspInESbasis_esCombinationcshInSchurBasis_schurCombinationmsPolynomialsInJackBasis msPolynomialsInJackSymbolicBasis