Îõ³h*UœSLÇ      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEF1.4.4.0 Safe-InferredÝ[GHIJKLMNOPQRSTUVWXYZ[\]^_`abcdEvaluation of Jack polynomials.(c) Stéphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-InferredÝjackpolynomialsEvaluation 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-Inferredµ jackpolynomialsÎInefficient hypergeometric function of a matrix argument (for testing purpose)  Symbolic Jack polynomials.(c) Stéphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-InferredÝñ jackpolynomialsSymbolic Jack polynomial jackpolynomialsSymbolic Jack polynomial jackpolynomialsSymbolic zonal polynomial jackpolynomialsSymbolic zonal polynomialjackpolynomialsSymbolic Schur polynomialjackpolynomialsSymbolic Schur polynomialjackpolynomialsSymbolic skew Schur polynomialjackpolynomialsSymbolic skew 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 variablesjackpolynomialspartition of integers jackpolynomialsnumber of variablesjackpolynomialspartition of integersjackpolynomialsnumber 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) Stéphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-InferredÝqjackpolynomials,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 Q$Some utilities for Jack polynomials.(c) Stéphane Laurent, 2024GPL-3laurent_step@outlook.fr Safe-InferredÃÝRâÃejackpolynomialsmonomial symmetric polynomialjackpolynomialsMonomial symmetric polynomial/putStrLn $ prettySpray' (msPolynomial 3 [2, 1])Ñ(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.Â-- 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 sprayjackpolynomialsÏSymmetric polynomial as a linear combination of monomial symmetric polynomials.jackpolynomialsÒPrints 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]jackpolynomialsÒPrints 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 f symmetric sprayjackpolynomialsàPrints a symmetric parametric spray as a linear combination of monomial symmetric polynomials.ÓputStrLn $ 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]jackpolynomialsçPrints a symmetric simple parametric spray as a linear combination of monomial symmetric polynomials.jackpolynomialsûLaplace-Beltrami operator on the space of homogeneous symmetric polynomials; neither symmetry and homogeneity are checked.jackpolynomialsýCalogero-Sutherland operator on the space of homogeneous symmetric polynomials; neither symmetry and homogeneity are checkedjackpolynomialsPower 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^3gjackpolynomialsÐmonomial symmetric polynomial as a linear combination of power sum polynomialshjackpolynomials$the factor in the Hall inner productijackpolynomialsÅsymmetric polynomial as a linear combination of power sum polynomialsjackpolynomialsáSymmetric polynomial as a linear combination of power sum polynomials. Symmetry is not checked. jackpolynomialsÑSymmetric polynomial as a linear combination of power sum polynomials. Same as  psCombination: but with other constraints on the base ring of the spray.jjackpolynomials%the Hall inner product with parameter!jackpolynomials‰Hall inner product with parameter, aka Jack-scalar product. It makes sense only for symmetric sprays, and the symmetry is not checked. "jackpolynomials+Hall inner product with parameter. Same as hallInnerProduct= but with other constraints on the base ring of the sprays.#jackpolynomials+Hall inner product with parameter. Same as hallInnerProductÔ but with other constraints on the base ring of the sprays. It is applicable to  Spray Int sprays.$jackpolynomialsßHall 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 RatioOfQSpraysÄ 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'ÏhallInnerProduct''' jp jp 5 == hallInnerProduct jp jp (constantRatioOfSprays 5)%jackpolynomialsÃHall inner product with parameter for parametric sprays. Same as hallInnerProduct'''À but with other constraints on the types. It is applicable to SimpleParametricQSpray sprays, while hallInnerProduct''' is not.kjackpolynomials.the Hall inner product with symbolic parameter&jackpolynomialsÉHall 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.)jackpolynomials)Complete symmetric homogeneous polynomial0putStrLn $ prettyQSpray (cshPolynomial 3 [2, 1])Õx^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^3ljackpolynomialsÜpower sum polynomial as a linear combination of complete symmetric homogeneous polynomialsmjackpolynomialsåmonomial symmetric polynomial as a linear combination of complete symmetric homogeneous polynomialsnjackpolynomialsÜsymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials*jackpolynomialsöSymmetric polynomial as a linear combination of complete symmetric homogeneous polynomials. Symmetry is not checked.+jackpolynomialsæSymmetric 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^2ojackpolynomialsÒpower sum polynomial as a linear combination of elementary symmetric polynomialspjackpolynomialsÛmonomial symmetric polynomial as a linear combination of elementary symmetric polynomialsqjackpolynomialsÒsymmetric polynomial as a linear combination of elementary symmetric polynomials-jackpolynomialsìSymmetric polynomial as a linear combination of elementary symmetric polynomials. Symmetry is not checked..jackpolynomialsÜSymmetric polynomial as a linear combination of elementary symmetric polynomials. Same as  esCombination: but with other constraints on the base ring of the spray.rjackpolynomialsØcomplete symmetric homogeneous polynomial as a linear combination of Schur polynomialssjackpolynomialsÁsymmetric polynomial as a linear combination of Schur polynomials/jackpolynomialsÝSymmetric polynomial as a linear combination of Schur polynomials. Symmetry is not checked.0jackpolynomialsÍSymmetric polynomial as a linear combination of Schur polynomials. Same as schurCombination: but with other constraints on the base ring of the spray.1jackpolynomialsKostka numbers K_{\lambda,\mu}(\alpha)( for a given weight of the partitions \lambda and \mu and a given 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 0.2jackpolynomialsKostka numbers K_{\lambda,\mu}(\alpha) with symbolic 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.tjackpolynomials8monomial symmetric polynomials in Jack polynomials basisujackpolynomials8monomial symmetric polynomials in Jack polynomials basis3jackpolynomialsøSymmetric polynomial as a linear combination of Jack polynomials with a given Jack parameter. Symmetry is not checked.4jackpolynomialsôSymmetric polynomial as a linear combination of Jack polynomials with symbolic parameter. Symmetry is not checked.5jackpolynomialsôSymmetric parametric polynomial as a linear combination of Jack polynomials with symbolic parameter. Similar to jackSymbolicCombination but for a parametric spray.6jackpolynomialsäKostka-Foulkes polynomial of two given partitions. This is a univariate polynomial whose value at 1, is the Kostka number of the two partitions.7jackpolynomialsäKostka-Foulkes polynomial of two given partitions. This is a univariate polynomial whose value at 1, is the Kostka number of the two partitions.8jackpolynomials‚Skew 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.9jackpolynomials‚Skew 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.:jackpolynomials’Hall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.;jackpolynomials’Hall-Littlewood polynomial of a given partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.<jackpolynomialsÈHall-Littlewood polynomials as linear combinations of Schur polynomials.=jackpolynomialsœSkew Hall-Littlewood polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.>jackpolynomialsœSkew Hall-Littlewood polynomial of a given skew partition. This is a multivariate symmetric polynomial whose coefficients are polynomial in one parameter.?jackpolynomialsÓFlagged Schur polynomial. A flagged Schur polynomial is not symmetric in general.@jackpolynomialsÓFlagged Schur polynomial. A flagged Schur polynomial is not symmetric in general.AjackpolynomialsÝFlagged skew Schur polynomial. A flagged skew Schur polynomial is not symmetric in general.BjackpolynomialsÝFlagged skew Schur polynomial. A flagged skew Schur polynomial is not symmetric in general.Cjackpolynomials!Factorial Schur polynomial. See  Êhttps://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r84/pdfKreiman's paper %Products of factorial Schur functions for the definition.Djackpolynomials!Factorial Schur polynomial. See  Êhttps://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r84/pdfKreiman's paper %Products of factorial Schur functions for the definition.Ejackpolynomials'Skew factorial Schur polynomial. See  Ëhttps://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/opapers/s28macdonald.pdfMacdonald's paper %Schur functions: theme and variations$, 6th variation, for the definition.Fjackpolynomials'Skew factorial Schur polynomial. See  Ëhttps://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/opapers/s28macdonald.pdfMacdonald's paper %Schur functions: theme and variations$, 6th variation, for the definition.ejackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjjackpolynomialssprayjackpolynomialssprayjackpolynomials parameter!jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter"jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter#jackpolynomialssprayjackpolynomialssprayjackpolynomials parameter$jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter%jackpolynomialsparametric sprayjackpolynomialsparametric sprayjackpolynomials parameter)jackpolynomialsnumber of variablesjackpolynomialsinteger partition,jackpolynomialsnumber of variablesjackpolynomialsinteger partition1jackpolynomialsweight of the partitionsjackpolynomialsJack parameter3jackpolynomialsJack 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 spray4jackpolynomialswhich 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 spray5jackpolynomialswhich 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 spray8jackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials›integer 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 polynomial9jackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials›integer 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 polynomial:jackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomials"which Hall-Littlewood polynomial, P or Q;jackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomials"which Hall-Littlewood polynomial, P or Q<jackpolynomials;weight of the partitions of the Hall-Littlewood polynomialsjackpolynomials#which Hall-Littlewood polynomials, P or Q=jackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials'which skew Hall-Littlewood polynomial, P or Q>jackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials'which skew Hall-Littlewood polynomial, P or Q?jackpolynomialsinteger partitionjackpolynomials lower boundsjackpolynomials upper bounds@jackpolynomialsinteger partitionjackpolynomials lower boundsjackpolynomials upper boundsAjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials lower boundsjackpolynomials upper boundsBjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomials lower boundsjackpolynomials upper boundsCjackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsthe sequence denoted by y in the reference paper Djackpolynomialsnumber of variablesjackpolynomialsinteger partitionjackpolynomialsthe sequence denoted by y in the reference paperEjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsthe sequence denoted by a in the reference paperFjackpolynomialsnumber of variablesjackpolynomials%outer partition of the skew partitionjackpolynomials%inner partition of the skew partitionjackpolynomialsthe sequence denoted by a in the reference paper3), *+-./0345!"#$%&'(126789:;<=>?@ABCDEF3), *+-./0345!"#$%&'(126789:;<=>?@ABCDEFö      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~€.jackpolynomials-1.4.4.0-DdLIwQFR1yfDS8sgfmol3HMath.Algebra.JackMath.Algebra.Jack.HypergeoPQMath.Algebra.JackPolMath.Algebra.JackSymbolicPol!Math.Algebra.SymmetricPolynomialsjackpolynomialsMath.Algebra.Jack.Internal Partitionjack'jackzonal'zonalschur'schur skewSchur' skewSchur hypergeoPQjackPol'jackPol zonalPol'zonalPol schurPol'schurPol skewSchurPol' skewSchurPoljackSymbolicPol'jackSymbolicPol msPolynomialisSymmetricSpray msCombinationprettySymmetricNumSprayprettySymmetricQSprayprettySymmetricQSpray'prettySymmetricParametricQSpray%prettySymmetricSimpleParametricQSpraylaplaceBeltramicalogeroSutherland psPolynomial psCombinationpsCombination'hallInnerProducthallInnerProduct'hallInnerProduct''hallInnerProduct'''hallInnerProduct''''symbolicHallInnerProductsymbolicHallInnerProduct'symbolicHallInnerProduct'' cshPolynomialcshCombinationcshCombination' esPolynomial esCombinationesCombination'schurCombinationschurCombination' kostkaNumberssymbolicKostkaNumbersjackCombinationjackSymbolicCombinationjackSymbolicCombination'kostkaFoulkesPolynomialkostkaFoulkesPolynomial'skewKostkaFoulkesPolynomialskewKostkaFoulkesPolynomial'hallLittlewoodPolynomialhallLittlewoodPolynomial' transitionsSchurToHallLittlewoodskewHallLittlewoodPolynomialskewHallLittlewoodPolynomial'flaggedSchurPolflaggedSchurPol'flaggedSkewSchurPolflaggedSkewSchurPol'factorialSchurPolfactorialSchurPol'skewFactorialSchurPolskewFactorialSchurPol' jackCoeffP jackCoeffQ jackCoeffCjackSymbolicCoeffCjackSymbolicCoeffPinvjackSymbolicCoeffQinv _betaratio_betaRatioOfSprays _isPartition_N_fromIntskewSchurLRCoefficientsisSkewPartition sprayToMap comboToSpray_kostkaNumbers_inverseKostkaMatrix_symbolicKostkaNumbers_inverseSymbolicKostkaMatrix_kostkaFoulkesPolynomial&_hallLittlewoodPolynomialsInSchurBasis$_transitionMatrixHallLittlewoodSchurskewHallLittlewoodPskewHallLittlewoodQ flaggedSemiStandardYoungTableaux tableauWeight isIncreasingflaggedSkewTableauxskewTableauWeight_skewKostkaFoulkesPolynomialmsPolynomialUnsafe%hspray-0.5.3.0-4VZscisOfrQJlwwwy5IVlaMath.Algebra.HsprayQSpray' mspInPSbasiszlambda_psCombination_hallInnerProduct_symbolicHallInnerProduct pspInCSHbasis mspInCSHbasis_cshCombination pspInESbasis mspInESbasis_esCombinationcshInSchurBasis_schurCombinationmsPolynomialsInJackBasis msPolynomialsInJackSymbolicBasis