[Merged by Bors] - feat(Combinatorics): generating function for partitions

[wwylele]

This is the start of a series of PR aiming to prove two theorems related to partition:

• Glaisher's theorem, which is a generalization of the existing Euler's Partition theorem. The proof will use the infinite generating function, upgrading the current proof that uses finite ones, and resolving this TODO
• Pentagonal number theorem and the recurrence relation on the partition function.

I created a new file Mathlib/Combinatorics/Enumerative/Partition/GenFun.lean to avoid importing PowerSeries stuff directly into the definition file, and along the way I moved the existing Mathlib/Combinatorics/Enumerative/Partition.lean to Mathlib/Combinatorics/Enumerative/Partition/Basic.lean .

---

• depends on: [Merged by Bors] - chore(Finsupp): move mapRange.equiv earlier #31324

[github-actions]

PR summary fe51f2299c

Import changes exceeding 2%

%	File
+2.37%	Mathlib.GroupTheory.Perm.Cycle.Type

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.GroupTheory.Perm.Cycle.Type	1057	1082	+25 (+2.37%)
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs	1080	1091	+11 (+1.02%)

Import changes for all files

Files	Import difference
Mathlib.Combinatorics.Enumerative.Partition	-723
21 files Mathlib.Analysis.Distribution.FourierSchwartz Mathlib.Analysis.Distribution.SchwartzSpace Mathlib.Analysis.Distribution.TemperateGrowth Mathlib.Analysis.Fourier.FourierTransformDeriv Mathlib.Analysis.Fourier.PoissonSummation Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation Mathlib.NumberTheory.EulerProduct.DirichletLSeries Mathlib.NumberTheory.Harmonic.ZetaAsymp Mathlib.NumberTheory.LSeries.DirichletContinuation Mathlib.NumberTheory.LSeries.Dirichlet Mathlib.NumberTheory.LSeries.HurwitzZetaEven Mathlib.NumberTheory.LSeries.HurwitzZetaOdd Mathlib.NumberTheory.LSeries.HurwitzZeta Mathlib.NumberTheory.LSeries.Nonvanishing Mathlib.NumberTheory.LSeries.PrimesInAP Mathlib.NumberTheory.LSeries.RiemannZeta Mathlib.NumberTheory.LSeries.ZMod Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds Mathlib.NumberTheory.ModularForms.JacobiTheta.Manifold Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable	1
Mathlib.Analysis.InnerProductSpace.StarOrder	3
8 files Mathlib.Algebra.Lie.Weights.IsSimple Mathlib.Algebra.Lie.Weights.RootSystem Mathlib.Analysis.Matrix.Order Mathlib.Analysis.Matrix.PosDef Mathlib.Combinatorics.SimpleGraph.LapMatrix Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple	4
53 files Mathlib.Algebra.Category.ModuleCat.ExteriorPower Mathlib.Analysis.Calculus.LineDeriv.QuadraticMap Mathlib.Analysis.InnerProductSpace.GramMatrix Mathlib.Analysis.InnerProductSpace.Positive Mathlib.Combinatorics.Extremal.RuzsaSzemeredi Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange Mathlib.LinearAlgebra.CliffordAlgebra.Basic Mathlib.LinearAlgebra.CliffordAlgebra.CategoryTheory Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation Mathlib.LinearAlgebra.CliffordAlgebra.Contraction Mathlib.LinearAlgebra.CliffordAlgebra.Equivs Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv Mathlib.LinearAlgebra.CliffordAlgebra.Even Mathlib.LinearAlgebra.CliffordAlgebra.Fold Mathlib.LinearAlgebra.CliffordAlgebra.Grading Mathlib.LinearAlgebra.CliffordAlgebra.Inversion Mathlib.LinearAlgebra.CliffordAlgebra.Prod Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup Mathlib.LinearAlgebra.CliffordAlgebra.Star Mathlib.LinearAlgebra.ExteriorAlgebra.Basic Mathlib.LinearAlgebra.ExteriorAlgebra.Grading Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating Mathlib.LinearAlgebra.ExteriorPower.Basic Mathlib.LinearAlgebra.ExteriorPower.Pairing Mathlib.LinearAlgebra.Matrix.LDL Mathlib.LinearAlgebra.Matrix.PosDef Mathlib.LinearAlgebra.QuadraticForm.Basic Mathlib.LinearAlgebra.QuadraticForm.Basis Mathlib.LinearAlgebra.QuadraticForm.Complex Mathlib.LinearAlgebra.QuadraticForm.Dual Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv Mathlib.LinearAlgebra.QuadraticForm.Isometry Mathlib.LinearAlgebra.QuadraticForm.Prod Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat.Monoidal Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat.Symmetric Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat Mathlib.LinearAlgebra.QuadraticForm.Real Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.Isometries Mathlib.LinearAlgebra.QuadraticForm.TensorProduct Mathlib.LinearAlgebra.RootSystem.Base Mathlib.LinearAlgebra.RootSystem.Basic Mathlib.LinearAlgebra.RootSystem.CartanMatrix Mathlib.LinearAlgebra.RootSystem.Chain Mathlib.LinearAlgebra.RootSystem.Finite.G2 Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas Mathlib.LinearAlgebra.RootSystem.Hom Mathlib.LinearAlgebra.RootSystem.Irreducible Mathlib.LinearAlgebra.RootSystem.RootPairingCat Mathlib.LinearAlgebra.RootSystem.WeylGroup Mathlib.LinearAlgebra.SesquilinearForm.Star Mathlib.Probability.Moments.CovarianceBilin	5
8 files Mathlib.AlgebraicGeometry.AffineSpace Mathlib.AlgebraicGeometry.Morphisms.Etale Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation Mathlib.AlgebraicGeometry.Morphisms.Smooth Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen Mathlib.AlgebraicGeometry.Sites.Etale Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity Mathlib.RingTheory.Spectrum.Prime.Chevalley	7
70 files Mathlib.Analysis.CStarAlgebra.ApproximateUnit Mathlib.Analysis.CStarAlgebra.CStarMatrix Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order Mathlib.Analysis.CStarAlgebra.GelfandDuality Mathlib.Analysis.CStarAlgebra.Hom Mathlib.Analysis.CStarAlgebra.Module.Constructions Mathlib.Analysis.CStarAlgebra.Module.Defs Mathlib.Analysis.CStarAlgebra.PositiveLinearMap Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart Mathlib.Analysis.CStarAlgebra.Spectrum Mathlib.Analysis.CStarAlgebra.Unitary.Connected Mathlib.Analysis.CStarAlgebra.Unitary.Span Mathlib.Analysis.Complex.Polynomial.Basic Mathlib.Analysis.Complex.Polynomial.GaussLucas Mathlib.Analysis.Complex.Polynomial.UnitTrinomial Mathlib.Analysis.Fourier.ZMod Mathlib.Analysis.Normed.Algebra.GelfandFormula Mathlib.Analysis.Normed.Algebra.GelfandMazur Mathlib.Analysis.Polynomial.Factorization Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation Mathlib.NumberTheory.Cyclotomic.Embeddings Mathlib.NumberTheory.Cyclotomic.PID Mathlib.NumberTheory.Cyclotomic.Three Mathlib.NumberTheory.DirichletCharacter.GaussSum Mathlib.NumberTheory.FLT.Three Mathlib.NumberTheory.Fermat Mathlib.NumberTheory.GaussSum Mathlib.NumberTheory.JacobiSum.Basic Mathlib.NumberTheory.LegendreSymbol.AddCharacter Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity Mathlib.NumberTheory.LucasLehmer Mathlib.NumberTheory.NumberField.AdeleRing Mathlib.NumberTheory.NumberField.CMField Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord Mathlib.NumberTheory.NumberField.ClassNumber Mathlib.NumberTheory.NumberField.Completion Mathlib.NumberTheory.NumberField.Cyclotomic.Embeddings Mathlib.NumberTheory.NumberField.Cyclotomic.PID Mathlib.NumberTheory.NumberField.Cyclotomic.Three Mathlib.NumberTheory.NumberField.DedekindZeta Mathlib.NumberTheory.NumberField.Discriminant.Basic Mathlib.NumberTheory.NumberField.Discriminant.Different Mathlib.NumberTheory.NumberField.Embeddings Mathlib.NumberTheory.NumberField.EquivReindex Mathlib.NumberTheory.NumberField.FinitePlaces Mathlib.NumberTheory.NumberField.House Mathlib.NumberTheory.NumberField.Ideal.Asymptotics Mathlib.NumberTheory.NumberField.Ideal.Basic Mathlib.NumberTheory.NumberField.Ideal Mathlib.NumberTheory.NumberField.InfinitePlace.Basic Mathlib.NumberTheory.NumberField.InfinitePlace.Completion Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex Mathlib.NumberTheory.NumberField.ProductFormula Mathlib.NumberTheory.NumberField.Units.Basic Mathlib.NumberTheory.NumberField.Units.DirichletTheorem Mathlib.NumberTheory.NumberField.Units.Regulator Mathlib.RingTheory.Polynomial.Selmer Mathlib.Tactic.NormNum.LegendreSymbol Mathlib.Tactic	8
439 files Mathlib.Algebra.Category.Ring.Under.Limits Mathlib.Algebra.Lie.Derivation.Killing Mathlib.Algebra.Lie.Killing Mathlib.Algebra.Lie.LieTheorem Mathlib.Algebra.Lie.Semisimple.Lemmas Mathlib.Algebra.Lie.TraceForm Mathlib.Algebra.Lie.Weights.Basic Mathlib.Algebra.Lie.Weights.Cartan Mathlib.Algebra.Lie.Weights.Chain Mathlib.Algebra.Lie.Weights.Killing Mathlib.Algebra.Lie.Weights.Linear Mathlib.Algebra.Module.Presentation.Differentials Mathlib.Algebra.Module.ZLattice.Covolume Mathlib.Algebra.Polynomial.Bivariate Mathlib.AlgebraicGeometry.AffineScheme Mathlib.AlgebraicGeometry.AffineTransitionLimit Mathlib.AlgebraicGeometry.Cover.Directed Mathlib.AlgebraicGeometry.Cover.MorphismProperty Mathlib.AlgebraicGeometry.Cover.Open Mathlib.AlgebraicGeometry.Cover.Over Mathlib.AlgebraicGeometry.Cover.Sigma Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point Mathlib.AlgebraicGeometry.EllipticCurve.Affine Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree Mathlib.AlgebraicGeometry.EllipticCurve.Group Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point Mathlib.AlgebraicGeometry.EllipticCurve.Reduction Mathlib.AlgebraicGeometry.Fiber Mathlib.AlgebraicGeometry.FunctionField Mathlib.AlgebraicGeometry.GammaSpecAdjunction Mathlib.AlgebraicGeometry.GluingOneHypercover Mathlib.AlgebraicGeometry.Gluing Mathlib.AlgebraicGeometry.IdealSheaf.Basic Mathlib.AlgebraicGeometry.IdealSheaf.Functorial Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme Mathlib.AlgebraicGeometry.IdealSheaf Mathlib.AlgebraicGeometry.Limits Mathlib.AlgebraicGeometry.Modules.Presheaf Mathlib.AlgebraicGeometry.Modules.Sheaf Mathlib.AlgebraicGeometry.Modules.Tilde Mathlib.AlgebraicGeometry.Morphisms.AffineAnd Mathlib.AlgebraicGeometry.Morphisms.Affine Mathlib.AlgebraicGeometry.Morphisms.Basic Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion Mathlib.AlgebraicGeometry.Morphisms.Constructors Mathlib.AlgebraicGeometry.Morphisms.Descent Mathlib.AlgebraicGeometry.Morphisms.FiniteType Mathlib.AlgebraicGeometry.Morphisms.Finite Mathlib.AlgebraicGeometry.Morphisms.Flat Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified Mathlib.AlgebraicGeometry.Morphisms.Immersion Mathlib.AlgebraicGeometry.Morphisms.Integral Mathlib.AlgebraicGeometry.Morphisms.IsIso Mathlib.AlgebraicGeometry.Morphisms.LocalClosure Mathlib.AlgebraicGeometry.Morphisms.LocalIso Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion Mathlib.AlgebraicGeometry.Morphisms.Preimmersion 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Mathlib.Combinatorics.Enumerative.Partition.Basic	771
Mathlib.Combinatorics.Enumerative.Partition.GenFun (new file)	1434

---

Declarations diff

+ coeff_genFun
+ finsuppAntidiag_mono
+ genFun
+ genFun_eq_tprod
+ hasProd_genFun
+ le_of_mem_parts
+ multipliable_genFun
+ summable_genFun_term
+ summable_genFun_term'
+ toFinsuppAntidiag
+ toFinsuppAntidiag_injective
+ toFinsuppAntidiag_mem_finsuppAntidiag

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

---

note: file Mathlib/Combinatorics/Enumerative/Partition.lean` was renamed to `Mathlib/Combinatorics/Enumerative/Partition/Basic.lean without a module deprecation
Please create a follow-up pull request adding one. Thanks!

[YaelDillies]

Not sure why github didn't recognize this as a file move, but there is no diff in that file and the total diff is much smaller

This is because you're adding the module deprecation simultaneously. Could you move the module deprecation to a later PR?

[wwylele]

This is because you're adding the module deprecation simultaneously. Could you move the module deprecation to a later PR?

Done! Thanks for the suggestion

[wwylele]

Pushed a large diff to remove the index shifting from toFinsuppAntidiag. In the end I moved all index shifting to hasProd_genFun itself and all the auxiliary lemmas now use natural indices. The end result made the overall code shorter 🥳

[wwylele]

Marked this as depending on #31324 as I like having the definition moved

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - chore(Finsupp): move mapRange.equiv earlier #31324
  By Dependent Issues (🤖). Happy coding!

[wwylele]

Rebased and moved the definition & lemma for toFinsuppAntidiag

[YaelDillies]

Thanks! 🚀

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by YaelDillies.

[b-mehta]

Thanks!

bors d+

[mathlib-bors]

✌️ wwylele can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

[wwylele]

bors r+

[mathlib-bors]

Pull request successfully merged into master.

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