h*      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcd1.4.6.0 Safe-Inferred-efghijklmnopqrstuvwxyz{|}~Evaluation of Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferred 2jackpolynomials Evaluation of a Jack polynomial.jackpolynomials Evaluation of a Jack polynomial.jackpolynomialsEvaluation of a zonal polynomial. The zonal polynomials are the Jack C!-polynomials with Jack parameter \alpha=2.jackpolynomialsEvaluation of a zonal polynomial. The zonal polynomials are the Jack C!-polynomials with Jack parameter \alpha=2.jackpolynomialsEvaluation of a Schur polynomial. The Schur polynomials are the Jack P!-polynomials with Jack parameter \alpha=1.jackpolynomialsEvaluation of a Schur polynomial. The Schur polynomials are the Jack P!-polynomials with Jack parameter \alpha=1.jackpolynomials%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 variablesjackpolynomialsinteger partition jackpolynomialsvalues of the variablesjackpolynomialspartition of integersjackpolynomialsvalues of the variablesjackpolynomialsinteger partition 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  jackpolynomialsInefficient hypergeometric function of a matrix argument (for testing purpose)  (c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferred jackpolynomialsKostka numbers K_{\lambda,\mu}(\alpha) with Jack parameter, or Kostka-Jack numbers, 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. The Kostka-Jack number K_{\lambda,\mu}(\alpha); is the coefficient of the monomial symmetric polynomial m_\mu in the expression of the P-Jack polynomial P_\lambda(\alpha)= as a linear combination of monomial symmetric polynomials. jackpolynomialsKostka 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. For \alpha=1 these are the ordinary skew Kostka numbers. The function returns a map whose keys represent the partitions \nu . The skew Kostka-Jack number K_{\lambda/\mu, \nu}(\alpha)< is the coefficient of the monomial symmetric polynomial m_\nu in the expression of the skew P-Jack polynomial P_{\lambda/\mu}(\alpha)= as a linear combination of monomial symmetric polynomials. 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.jackpolynomialsSkew Gelfand-Tsetlin patterns defined by a skew partition and a weight vector.jackpolynomialsSkew semistandard tableaux with a given shape (a skew partition) and a given weight vector. The weight is the vector whose i.-th element is the number of occurrences of i in the tableau.jackpolynomialsSemistandard tableaux with a given shape (an integer partition) and a given weight vector. The weight is the vector whose i.-th element is the number of occurrences of i in the tableau. jackpolynomialsweight of the partitionsjackpolynomialsJack parameter jackpolynomialsweight of the partitions 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 partitionjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsweightjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsweightjackpolynomialsshape, integer partitionjackpolynomialsweight  Symbolic Jack polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferred$z jackpolynomialsJack polynomial.jackpolynomialsJack polynomial.jackpolynomialsSkew Jack polynomial.jackpolynomialsSkew Jack polynomial.jackpolynomials7Zonal polynomial. The zonal polynomials are the Jack C!-polynomials with Jack parameter \alpha=2.jackpolynomials7Zonal polynomial. The zonal polynomials are the Jack C!-polynomials with Jack parameter \alpha=2.jackpolynomialsSkew zonal polynomial.jackpolynomialsSkew zonal polynomial.jackpolynomials7Schur polynomial. The Schur polynomials are the Jack P!-polynomials with Jack parameter \alpha=1.jackpolynomials7Schur polynomial. The Schur polynomials are the Jack P!-polynomials with Jack parameter \alpha=1.jackpolynomialsSkew Schur polynomialjackpolynomialsSkew Schur polynomial jackpolynomialsnumber of variablesjackpolynomialsinteger partition jackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomialsinteger partition jackpolynomialsJack parameterjackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsJack parameterjackpolynomialswhich skew Jack polynomial, J, C, P or Qjackpolynomialsnumber 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-Inferred)jackpolynomials/Jack polynomial with a symbolic Jack parameter.jackpolynomials/Jack polynomial with a symbolic Jack parameter.jackpolynomials4Skew Jack polynomial with a symbolic Jack parameter. jackpolynomials4Skew Jack polynomial with a symbolic Jack parameter.jackpolynomialsnumber of variablesjackpolynomialsinteger partition jackpolynomialswhich Jack polynomial, J, C, P or Qjackpolynomialsnumber of variablesjackpolynomialsinteger partition jackpolynomialswhich 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 jackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialswhich skew Jack polynomial, J, C, P or Q  More symmetric polynomials.(c) Stphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-Inferred8jackpolynomialsmonomial symmetric polynomial!jackpolynomialsMonomial 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.x3^2"jackpolynomials6Checks 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 spray#jackpolynomialsSymmetric 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. 0jackpolynomials0Hall inner product with Jack parameter. Same as hallInnerProduct= but with other constraints on the base ring of the sprays.1jackpolynomials0Hall inner product with Jack parameter. Same as hallInnerProduct but with other constraints on the base ring of the sprays. It is applicable to  Spray Int sprays.2jackpolynomialsHall inner product with Jack 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)3jackpolynomialsHall inner product with Jack 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 parameter4jackpolynomialsHall inner product with symbolic Jack parameter. See README for some examples.5jackpolynomials9Hall inner product with symbolic Jack parameter. Same as symbolicHallInnerProduct# but with other type constraints.6jackpolynomials9Hall inner product with symbolic Jack parameter. Same as symbolicHallInnerProduct8 but with other type constraints. It is applicable to  Spray Int sprays.7jackpolynomials)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 polynomials8jackpolynomialsSymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials. Symmetry is not checked.9jackpolynomialsSymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials. Same as cshCombination: but with other constraints on the base ring of the spray.:jackpolynomials 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 polynomials;jackpolynomialsSymmetric polynomial as a linear combination of elementary symmetric polynomials. Symmetry is not checked.<jackpolynomialsSymmetric 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 polynomials=jackpolynomialsSymmetric polynomial as a linear combination of Schur polynomials. Symmetry is not checked.>jackpolynomialsSymmetric polynomial as a linear combination of Schur polynomials. Same as schurCombination: but with other constraints on the base ring of the spray.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.AjackpolynomialsSymmetric parametric polynomial as a linear combination of Jack polynomials with symbolic parameter. Similar to jackSymbolicCombination but for a parametric spray.BjackpolynomialsKostka-Foulkes polynomial of two given partitions. This is a univariate polynomial whose value at 1, is the Kostka number of the two partitions.CjackpolynomialsKostka-Foulkes polynomial of two given partitions. This is a univariate polynomial whose value at 1, is the Kostka number of the two partitions.DjackpolynomialsSkew 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.EjackpolynomialsSkew 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.Fjackpolynomialsqt-Kostka polynomials, aka Kostka-Macdonald polynomials. These are bivariate symmetric polynomials usually denoted by K_{\lambda, \mu}(q,t) for two integer partitions \lambda and mu, and q and t denote the variables. One obtains the Kostka-Foulkes polynomials by substituting q with 0. For a given partition \mu(, the function returns the polynomials K_{\lambda, \mu}(q,t) for all partitions \lambda of the same weight as \mu.Gjackpolynomialsqt-Kostka polynomials, aka Kostka-Macdonald polynomials. These are bivariate symmetric polynomials usually denoted by K_{\lambda, \mu}(q,t) for two integer partitions \lambda and mu, and q and t denote the variables. One obtains the Kostka-Foulkes polynomials by substituting q with 0. For a given partition \mu(, the function returns the polynomials K_{\lambda, \mu}(q,t) for all partitions \lambda of the same weight as \mu.HjackpolynomialsSkew qt-Kostka polynomials. These are bivariate symmetric polynomials usually denoted by K_{\lambda/\mu, \nu}(q,t) for two integer partitions \lambda and mu3 defining a skew partition, an integer partition \nu, and q and t denote the variables. One obtains the skew Kostka-Foulkes polynomials by substituting q with 0. For given partitions \lambda and \mu), the function returns the polynomials K_{\lambda/\mu, \nu}(q,t) for all partitions \nu, of the same weight as the skew partition.IjackpolynomialsSkew qt-Kostka polynomials. These are bivariate symmetric polynomials usually denoted by K_{\lambda/\mu, \nu}(q,t) for two integer partitions \lambda and mu3 defining a skew partition, an integer partition \nu, and q and t denote the variables. One obtains the skew Kostka-Foulkes polynomials by substituting q with 0. For given partitions \lambda and \mu), the function returns the polynomials K_{\lambda/\mu, \nu}(q,t) for all partitions \nu, of the same weight as the skew partition.JjackpolynomialsHall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t. When substituting t with 0 in the Hall-Littlewood P0-polynomials, one obtains the Schur polynomials.KjackpolynomialsHall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t. When substituting t with 0 in the Hall-Littlewood P0-polynomials, one obtains the Schur polynomials.LjackpolynomialsHall-Littlewood polynomials as linear combinations of Schur polynomials.MjackpolynomialsSkew Hall-Littlewood polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t. When substituting t with 0 in the skew Hall-Littlewood P5-polynomials, one obtains the skew Schur polynomials.NjackpolynomialsSkew Hall-Littlewood polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t. When substituting t with 0 in the skew Hall-Littlewood P5-polynomials, one obtains the skew Schur polynomials.Ojackpolynomialst-Schur polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t5. One obtains the Schur polynomials by substituting t with 0. Pjackpolynomialst-Schur polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t5. One obtains the Schur polynomials by substituting t with 0. QjackpolynomialsSkew t-Schur polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t;. One obtains the skew Schur polynomials by substituting t with 0. RjackpolynomialsSkew t-Schur polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in a single parameter usually denoted by t;. One obtains the skew Schur polynomials by substituting t with 0. SjackpolynomialsMacdonald polynomial. This is a symmetric multivariate polynomial depending on two parameters usually denoted by q and t. Substituting q with 0( yields the Hall-Littlewood polynomials.*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]TjackpolynomialsMacdonald polynomial. This is a symmetric multivariate polynomial depending on two parameters usually denoted by q and t. Substituting q with 0( yields the Hall-Littlewood polynomials.UjackpolynomialsSkew Macdonald polynomial of a given skew partition. This is a multivariate symmetric polynomial with two parameters usually denoted by q and t. Substituting q with 0- yields the skew Hall-Littlewood polynomials.VjackpolynomialsSkew Macdonald polynomial of a given skew partition. This is a multivariate symmetric polynomial with two parameters usually denoted by q and t. Substituting q with 0- yields the skew Hall-Littlewood polynomials.WjackpolynomialsMacdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in two parameters.XjackpolynomialsMacdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomial in two parameters.YjackpolynomialsSkew Macdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients depend on two parameters.ZjackpolynomialsSkew Macdonald J-polynomial. This is a multivariate symmetric polynomial whose coefficients depend on two parameters.[jackpolynomialsModified Macdonald polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomials in two parameters.\jackpolynomialsModified Macdonald polynomial. This is a multivariate symmetric polynomial whose coefficients are polynomials in two parameters.]jackpolynomialsFlagged 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.ajackpolynomials!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.bjackpolynomials!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.cjackpolynomials'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$, 6th variation, for the definition.djackpolynomials'Skew factorial Schur polynomial. See  https://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/opapers/s28macdonald.pdfMacdonald's paper %Schur functions: theme and variations$, 6th variation, for the definition..jackpolynomialsnumber of variablesjackpolynomialsinteger partition!jackpolynomialsnumber of variablesjackpolynomialsinteger partition+jackpolynomialsnumber of variablesjackpolynomialsinteger partition.jackpolynomialsparametric sprayjackpolynomialssprayjackpolynomialssprayjackpolynomials parameter/jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter0jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter1jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter2jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter3jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter7jackpolynomialsnumber of variablesjackpolynomialsinteger partition:jackpolynomialsnumber of variablesjackpolynomialsinteger 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 sprayAjackpolynomialswhich 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 sprayDjackpolynomials%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 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 polynomialHjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionIjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionJjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomials"which Hall-Littlewood polynomial, P or QKjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomials"which Hall-Littlewood polynomial, P or QLjackpolynomials;weight of the partitions of the Hall-Littlewood polynomialsjackpolynomials#which Hall-Littlewood polynomials, P or QMjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials'which skew Hall-Littlewood polynomial, P or QNjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials'which skew Hall-Littlewood polynomial, P or QOjackpolynomialsnumber of variablesjackpolynomialsinteger partitionPjackpolynomialsnumber of variablesjackpolynomialsinteger partitionQjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionRjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionSjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialswhich Macdonald polynomial, P or QTjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialswhich Macdonald polynomial, P or QUjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials!which skew Macdonald polynomial, P or QVjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials!which skew Macdonald polynomial, P or QWjackpolynomialsnumber of variablesjackpolynomialsinteger partitionXjackpolynomialsnumber of variablesjackpolynomialsinteger partitionYjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionZjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partition[jackpolynomialsnumber of variablesjackpolynomialsinteger partition\jackpolynomialsnumber of variablesjackpolynomialsinteger partition]jackpolynomialsinteger 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 boundsajackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsthe sequence denoted by y in the reference paper bjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsthe sequence denoted by y in the reference papercjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsthe sequence denoted by a in the reference paperdjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsthe sequence denoted by a in the reference paper"!+7:#,-.89;<=>?@A$%&'()*/0123456BCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcd"!+7:#,-.89;<=>?@A$%&'()*/0123456BCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcd      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~.jackpolynomials-1.4.6.0-6bcj1v0eFrw1EneNAAmzvnMath.Algebra.JackMath.Algebra.Jack.HypergeoPQMath.Algebra.CombinatoricsMath.Algebra.JackPolMath.Algebra.JackSymbolicPol!Math.Algebra.SymmetricPolynomialsjackpolynomialsMath.Algebra.Jack.Internal Partitionjack'jackzonal'zonalschur'schur 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