{ "chapters": [ { "id": "ac-notations-and-elementary-facts-examples", "title": "Notations and elementary facts + Examples", "source_path": "AlgebraicCombinatorics/tex/FPS/Notations.tex", "target_theorems": [ "def.binom.binom", "prop.binom.rec", "prop.binom.0", "thm.binom.sym" ] }, { "id": "ac-definitions-preamble-reminder-commutativ", "title": "Definitions: (preamble) ... Reminder: Commutative rings", "source_path": "AlgebraicCombinatorics/tex/FPS/CommutativeRings.tex", "target_theorems": [ "def.alg.commring", "def.alg.module" ] }, { "id": "ac-definitions-the-definition-of-formal-pow", "title": "Definitions: The definition of formal power ... What next?", "source_path": "AlgebraicCombinatorics/tex/FPS/FPSDefinition.tex", "target_theorems": [ "def.fps.fps", "def.fps.ops", "thm.fps.ring", "def.fps.coeff", "def.infsum.essfin", "def.fps.summable", "prop.fps.summable.sub", "prop.fps.summable-sums-rule", "def.fps.x", "lem.fps.xa", "prop.fps.xk", "cor.fps.sumakxk", "prop.binom.vandermonde.NN", "thm.binom.vandermonde.CC" ] }, { "id": "ac-dividing-fpss", "title": "Dividing FPSs", "source_path": "AlgebraicCombinatorics/tex/FPS/DividingFPS.tex", "target_theorems": [ "def.commring.inverse", "thm.commring.inverse-uni", "def.commring.fracs", "prop.commring.fracs.1", "prop.fps.invertible", "cor.fps.invertible.field", "prop.fps.invertible.1+x", "thm.fps.newton-binom", "thm.binom.upneg-n", "prop.fps.anti-newton-binom", "cor.fps.anti-newton-binom-2", "def.fps.div-by-x", "prop.fps.div-by-x-inverts", "lem.fps.g=xh", "lem.fps.first-n-coeffs-of-xna", "lem.fps.muls-of-xn", "lem.fps.prod.irlv.fg", "lem.fps.prod.irlv.mul", "lem.fps.prod.irlv.cong-mul" ] }, { "id": "ac-polynomials", "title": "Polynomials", "source_path": "AlgebraicCombinatorics/tex/FPS/Polynomials.tex", "target_theorems": [ "def.fps.pol", "thm.fps.pol.ring", "def.alg.ring", "def.alg.Kalg", "def.pol.subs", "thm.pol.eval.a+b" ] }, { "id": "ac-substitution-and-evaluation-of-power-ser", "title": "Substitution and evaluation of power series", "source_path": "AlgebraicCombinatorics/tex/FPS/Substitution.tex", "target_theorems": [ "def.fps.subs", "prop.fps.subs.wd", "prop.fps.subs.rules", "lem.fps.fg-coeffs-0", "def.kron-delta" ] }, { "id": "ac-derivatives-of-fpss", "title": "Derivatives of FPSs", "source_path": "AlgebraicCombinatorics/tex/FPS/Derivatives.tex", "target_theorems": [ "def.fps.deriv", "thm.fps.deriv.rules" ] }, { "id": "ac-exponentials-and-logarithms", "title": "Exponentials and logarithms", "source_path": "AlgebraicCombinatorics/tex/FPS/ExpLog.tex", "target_theorems": [ "def.fps.exp-log", "prop.fps.exp-log-der", "lem.fps.compos-cst-term-0", "thm.fps.exp-log-inv", "def.fps.Exp-Log-maps", "lem.fps.Exp-Log-maps-wd", "lem.fps.Exp-Log-maps-inv", "lem.fps.Exp-Log-additive", "prop.fps.Exp-Log-groups", "thm.fps.Exp-Log-group-iso", "def.fps.loder.1", "prop.fps.loder.log", "prop.fps.loder.prod", "cor.fps.loder.prodk", "cor.fps.loder.inv" ] }, { "id": "ac-non-integer-powers", "title": "Non-integer powers", "source_path": "AlgebraicCombinatorics/tex/FPS/NonIntegerPowers.tex", "target_theorems": [ "def.fps.power-c", "thm.fps.power-c.rules", "thm.fps.gen-newton", "prop.binom.nCk-2i-qedmo.CN" ] }, { "id": "ac-integer-compositions", "title": "Integer compositions", "source_path": "AlgebraicCombinatorics/tex/FPS/IntegerCompositions.tex", "target_theorems": [ "def.fps.comps", "thm.fps.comps.num-comps-n-k", "thm.fps.comps.num-comps-n", "def.fps.wcomps", "thm.fps.comps.num-wcomps-n-k", "thm.fps.comps.num-wpcomps-n-k", "prop.fps.comps.num-w2comps-n-k-id" ] }, { "id": "ac-x-n-equivalence", "title": "x^{n}-equivalence", "source_path": "AlgebraicCombinatorics/tex/FPS/XnEquivalence.tex", "target_theorems": [ "def.fps.xneq", "thm.fps.xneq.props", "prop.fps.xneq-multiple", "prop.fps.xneq.comp" ] }, { "id": "ac-infinite-products-preamble-properties-of", "title": "Infinite products: (preamble) ... Properties of infinite products", "source_path": "AlgebraicCombinatorics/tex/FPS/InfiniteProducts1.tex", "target_theorems": [ "def.fps.determines-xn-coeff", "def.fps.xn-coeff-fin-determined", "prop.fps.summable=fin-det", "def.fps.multipliable", "prop.fps.multipliable.prod-wd", "prop.fps.multipliable.prod-wd2", "lem.fps.prod.irlv.1", "lem.fps.prod.irlv.fin", "thm.fps.1+f-mulable", "prop.fps.1-mulable", "def.fps.infprod-approx", "lem.fps.mulable.approx", "prop.fps.infprod-approx-xneq", "prop.fps.union-mulable", "prop.fps.prod-mulable", "prop.fps.div-mulable", "prop.fps.prods-mulable-subfams", "prop.fps.prods-mulable-rules.reindex", "prop.fps.prods-mulable-rules.SW1", "prop.fps.prods-mulable-rules.fubini1", "prop.fps.prods-mulable-rules.fubini" ] }, { "id": "ac-infinite-products-product-rules-generali", "title": "Infinite products: Product rules (generalized distributive laws) ... Exponentials, logarithms and infinite products", "source_path": "AlgebraicCombinatorics/tex/FPS/InfiniteProducts2.tex", "target_theorems": [ "prop.fps.prodrule-fin-fin", "prop.fps.prodrule-fin-inf", "def.fps.prodrule.ess-fin", "prop.fps.prodrule-inf-infN", "prop.fps.prodrule-inf-inf", "prop.fps.prodrule-fin-infJ", "lem.fps.prod.irlv.inf", "prop.gf.prod.euler-odd", "thm.gf.prod.euler-comb", "prop.fps.subs.rule-infprod", "prop.fps.Exp-Log-infsum", "prop.fps.Exp-Log-infprod" ] }, { "id": "ac-the-generating-function-of-a-weighted", "title": "The generating function of a weighted", "source_path": "AlgebraicCombinatorics/tex/FPS/WeightedSets.tex", "target_theorems": [ "def.gf-ws.weighted-sets", "prop.gf-ws.iso", "def.gf-ws.djun", "prop.gf-ws.djun", "def.gf-ws.prod", "prop.gf-ws.prod", "prop.gf-ws.pow", "def.domino.shapes-and-tilings", "lem.gf.weighted-set.domino.fd" ] }, { "id": "ac-limits-of-fpss", "title": "Limits of FPSs", "source_path": "AlgebraicCombinatorics/tex/FPS/Limits.tex", "target_theorems": [ "def.fps.lim.stab", "def.fps.lim.coeff-stab", "thm.fps.lim.lim-crit", "lem.fps.lim.xn-equiv", "prop.fps.lim.sum-prod", "cor.fps.lim.sum-prod-k", "prop.fps.lim.sum-quot", "prop.fps.lim.comp", "prop.fps.lim.deriv-lim", "thm.fps.lim.sum-lim", "thm.fps.lim.prod-lim", "cor.fps.lim.fps-as-pol", "thm.fps.lim.sum-lim-conv", "thm.fps.lim.prod-lim-conv" ] }, { "id": "ac-laurent-power-series", "title": "Laurent power series", "source_path": "AlgebraicCombinatorics/tex/FPS/LaurentSeries.tex", "target_theorems": [ "thm.fps.laure.binary-rep-uniq", "thm.fps.laure.balanced-tern-rep-uniq", "def.fps.laure.double", "def.fps.laure.laupol", "thm.fps.laure.laupol-ring", "prop.fps.laure.a=sumaixi", "def.fps.laure.lauser", "thm.fps.laure.lauser-ring" ] }, { "id": "ac-multivariate-fpss", "title": "Multivariate FPSs", "source_path": "AlgebraicCombinatorics/tex/FPS/Multivariate.tex", "target_theorems": [ "prop.fps.mulvar.comp-y-coeff" ] }, { "id": "ac-partition-basics", "title": "Partition basics", "source_path": "AlgebraicCombinatorics/tex/Partitions/Basics.tex", "target_theorems": [ "def.pars.parts", "def.pars.pn-pkn", "def.pars.iverson", "def.pars.floor-ceil", "prop.pars.basics", "thm.pars.main-gf", "thm.pars.main-gf-parts-n", "thm.pars.main-gf-parts-I", "def.pars.odd-dist-parts", "thm.pars.odd-dist-equal", "prop.pars.pkn=dual", "cor.pars.p0kn=dual", "thm.pars.main-gf-0n", "thm.pars.sigma1", "thm.pars.sigma1-I" ] }, { "id": "ac-euler-s-pentagonal-number-theorem-jacobi", "title": "Euler's pentagonal number theorem + Jacobi's triple product identity", "source_path": "AlgebraicCombinatorics/tex/Partitions/PentagonalJacobi.tex", "target_theorems": [ "def.pars.pent-num", "thm.pars.pent", "cor.pars.pn-rec", "thm.pars.jtp1", "thm.pars.jtp2", "lem.fps.fxx=gxx", "thm.pars.euler-sum-div-rec" ] }, { "id": "ac-q-binomial-coefficients-preamble-basic-p", "title": "q-binomial coefficients: (preamble) ... Basic properties", "source_path": "AlgebraicCombinatorics/tex/Partitions/QBinomialBasic.tex", "target_theorems": [ "prop.pars.qbinom.intro-count-binom", "def.pars.qbinom.qbinom", "prop.pars.qbinom.alt-defs", "prop.pars.qbinom.0", "prop.pars.qbinom.n0", "thm.pars.qbinom.rec", "thm.pars.qbinom.quot1", "def.pars.qbinom.qint", "thm.pars.qbinom.quot2", "prop.pars.qbinom.symm" ] }, { "id": "ac-q-binomial-coefficients-q-binomial-formu", "title": "q-binomial coefficients: q-binomial formulas ... Limits of q-binomial coefficients", "source_path": "AlgebraicCombinatorics/tex/Partitions/QBinomialFormulas.tex", "target_theorems": [ "thm.pars.qbinom.binom1", "lem.prodrule.sum-ai-plus-bi", "thm.pars.qbinom.binom2", "thm.pars.qbinom.subsp-count", "lem.linalg.lin-ind-via-span", "lem.pars.qbinom.lin-ind-count", "lem.count.multijection", "prop.pars.qbinom.lim1" ] }, { "id": "ac-basic-definitions-transpositions-cycles", "title": "Basic definitions + Transpositions, cycles and involutions", "source_path": "AlgebraicCombinatorics/tex/Permutations/Basics.tex", "target_theorems": [ "def.perm.perm", "def.perm.Sn-iven", "prop.perm.Sf", "def.perm.notations", "def.perm.tij", "def.perm.si", "prop.perm.si.rules", "def.perm.cycs", "def.perm.invol" ] }, { "id": "ac-inversions-length-and-lehmer-codes-pream", "title": "Inversions, length and Lehmer codes: (preamble) ... Lehmer codes", "source_path": "AlgebraicCombinatorics/tex/Permutations/Inversions1.tex", "target_theorems": [ "def.perm.invs", "prop.perm.lengths-k-small-k", "prop.perm.length.gf", "def.perm.lehmer1", "prop.perm.lehmer.l", "thm.perm.lehmer.bij", "def.perm.lehmer.lex-ord", "prop.perm.lehmer.lex-ord.total", "prop.perm.lehmer.lex" ] }, { "id": "ac-inversions-length-and-lehmer-codes-more", "title": "Inversions, length and Lehmer codes: More about lengths and simples", "source_path": "AlgebraicCombinatorics/tex/Permutations/Inversions2.tex", "target_theorems": [ "prop.perm.len.inv", "lem.perm.len.ssl", "prop.perm.lisitij", "thm.perm.len.redword1", "cor.perm.red.sigtau", "cor.perm.generated", "prop.perm.redword-lehmer" ] }, { "id": "ac-signs-of-permutations", "title": "Signs of permutations", "source_path": "AlgebraicCombinatorics/tex/Permutations/Signs.tex", "target_theorems": [ "def.perm.sign", "prop.perm.sign.props", "cor.perm.sign.hom", "def.perm.even-odd", "cor.perm.altgp", "cor.perm.num-even", "prop.perm.sign.X" ] }, { "id": "ac-the-cycle-decomposition", "title": "The cycle decomposition", "source_path": "AlgebraicCombinatorics/tex/Permutations/CycleDecomposition.tex", "target_theorems": [ "thm.perm.dcd.main", "def.perm.cycs.cycs", "prop.perm.cycs.same", "prop.perm.cycs.sign" ] }, { "id": "ac-cancellations-in-alternating-sums", "title": "Cancellations in alternating sums", "source_path": "AlgebraicCombinatorics/tex/SignedCounting/AlternatingSums.tex", "target_theorems": [ "prop.binom.nhs", "lem.sign.cancel1", "lem.sign.cancel2", "lem.sign.cancel3", "def.root-of-unity.prim", "thm.sign.q-lucas" ] }, { "id": "ac-the-principles-of-inclusion-and-exclusio", "title": "The principles of inclusion and exclusion: (preamble) ... The weighted version", "source_path": "AlgebraicCombinatorics/tex/SignedCounting/InclusionExclusion1.tex", "target_theorems": [ "thm.pie.1", "thm.pie.count-sur", "cor.pie.count-sur.cors", "def.pie.dera", "thm.pie.count-der", "thm.pie.euler-tot", "thm.pie.2" ] }, { "id": "ac-the-principles-of-inclusion-and-exclusio-28", "title": "The principles of inclusion and exclusion: Boolean M\\\"{o}bius inversion", "source_path": "AlgebraicCombinatorics/tex/SignedCounting/InclusionExclusion2.tex", "target_theorems": [ "thm.pie.moeb", "lem.pie.two-sets-altsum" ] }, { "id": "ac-more-subtractive-methods", "title": "More subtractive methods", "source_path": "AlgebraicCombinatorics/tex/SignedCounting/SubtractiveMethods.tex", "target_theorems": [ "thm.cancel.all-even", "lem.cancel.all-even.l1", "lem.cancel.all-even.l2" ] }, { "id": "ac-determinants-preamble-basic-properties", "title": "Determinants: (preamble) ... Basic properties", "source_path": "AlgebraicCombinatorics/tex/Determinants/BasicProperties.tex", "target_theorems": [ "def.det.det", "prop.det.xiyj", "prop.det.xi+yj", "thm.det.transp", "thm.det.triang", "thm.det.rowop", "thm.det.colop", "cor.det.sig-row-col", "thm.det.detAB", "cor.det.scale-row-col" ] }, { "id": "ac-determinants-cauchy-binet-factoring-the", "title": "Determinants: Cauchy--Binet ... Factoring the matrix", "source_path": "AlgebraicCombinatorics/tex/Determinants/CauchyBinet.tex", "target_theorems": [ "thm.det.CB", "def.det.sub", "thm.det.det(A+B)", "lem.det.minors-diag", "thm.det.det(A+D)", "prop.det.x+ai", "prop.det.charpol-explicit", "prop.det.pascal-LU" ] }, { "id": "ac-determinants-factor-hunting-desnanot-jac", "title": "Determinants: Factor hunting ... Desnanot--Jacobi and Dodgson condensation", "source_path": "AlgebraicCombinatorics/tex/Determinants/DesnanotJacobi.tex", "target_theorems": [ "thm.det.vander", "lem.det.vander.a.pol", "prop.det.(xi+yj)n-1", "thm.det.laplace", "prop.det.laplace.0", "def.det.adj", "thm.det.adj.inverse", "thm.det.laplace-multi", "thm.det.des-jac-1", "thm.det.cauchy", "thm.det.des-jac-2", "thm.det.jacobi-complement" ] }, { "id": "ac-the-lindstrom-gessel-viennot-lemma-pream", "title": "The Lindstr{o}m--Gessel--Viennot lemma: (preamble) ... The LGV lemma for k paths", "source_path": "AlgebraicCombinatorics/tex/Determinants/LGV1.tex", "target_theorems": [ "def.lgv.lattice", "prop.lgv.1-paths.ct", "def.lgv.path-tups", "prop.lgv.2paths.count", "prop.lgv.jordan-2", "cor.lgv.binom-unimod", "prop.lgv.kpaths.count" ] }, { "id": "ac-the-lindstrom-gessel-viennot-lemma-the-w", "title": "The Lindstr{o}m--Gessel--Viennot lemma: The weighted version ... The nonpermutable case", "source_path": "AlgebraicCombinatorics/tex/Determinants/LGV2.tex", "target_theorems": [ "thm.lgv.kpaths.wt", "thm.lgv.kpaths.wt-dg", "cor.lgv.kpaths.wt-np", "cor.lgv.binom-det-nonneg", "cor.lgv.catalan-hankel-det-0" ] }, { "id": "ac-definitions-and-examples-of-symmetric", "title": "Definitions and examples of symmetric", "source_path": "AlgebraicCombinatorics/tex/SymmetricFunctions/Definitions.tex", "target_theorems": [ "def.sf.PS", "prop.sf.SN-acts", "prop.sf.SN-acts-by-alg-auts", "thm.sf.S-subalg", "def.sf.ring-of-symm", "def.sf.monomial", "def.sf.ehp", "prop.sf.en=0", "thm.sf.NG", "prop.sf.e-h-FPS", "thm.sf.ftsf", "lem.sf.simples-enough" ] }, { "id": "ac-n-partitions-and-monomial-symmetric-poly", "title": "N-partitions and monomial symmetric polynomials", "source_path": "AlgebraicCombinatorics/tex/SymmetricFunctions/MonomialSymmetric.tex", "target_theorems": [ "def.sf.Npar", "prop.sf.Npar-as-par", "def.sf.sort", "def.sf.m", "prop.sf.ehp-through-m", "thm.sf.m-basis", "prop.sf.sigma-pol-coeff" ] }, { "id": "ac-schur-polynomials-preamble-skew-young-di", "title": "Schur polynomials: (preamble) ... Skew Young diagrams and skew Schur polynomials", "source_path": "AlgebraicCombinatorics/tex/SymmetricFunctions/SchurBasics.tex", "target_theorems": [ "def.sf.alternants", "def.sf.ydiag", "def.sf.ytab", "def.sf.ssyt", "def.sf.ytab.xT", "def.sf.schur", "thm.sf.schur-symm", "def.sf.par-subset", "def.sf.skew-diag", "lem.sf.skew-diag.convexity", "def.sf.skew-tab", "def.sf.skew-ssyt", "lem.sf.skew-ssyt.increase", "def.sf.ytab.skew-xT", "def.sf.skew-schur", "thm.sf.skew-schur-symm" ] }, { "id": "ac-schur-polynomials-the-littlewood-richard", "title": "Schur polynomials: The Littlewood--Richardson rule (part 1)", "source_path": "AlgebraicCombinatorics/tex/SymmetricFunctions/LittlewoodRichardson.tex", "target_theorems": [ "def.sf.tuple-addition", "def.sf.content", "def.sf.col-tab", "def.sf.yamanouchi", "thm.sf.lr-zy", "lem.sf.stemb-lem", "lem.sf.tab-greater-i", "def.cring.reg", "lem.cring.reg.cancel", "lem.sf.arho-reg", "lem.sf.alternant-0" ] }, { "id": "ac-schur-polynomials-the-pieri-rules-the-ja", "title": "Schur polynomials: The Pieri rules ... The Jacobi--Trudi identities", "source_path": "AlgebraicCombinatorics/tex/SymmetricFunctions/PieriJacobiTrudi.tex", "target_theorems": [ "def.sf.strips", "prop.sf.strips.entries", "thm.sf.pieri", "thm.sf.jt-h", "thm.sf.jt-e" ] }, { "id": "ac-details-infinite-products-part-1-part-2", "title": "Details: Infinite products (part 1) (part 2)", "source_path": "AlgebraicCombinatorics/tex/Details/InfiniteProducts1.tex", "target_theorems": [ "lem.fps.prod.irlv.cong-div", "lem.fps.prods-mulable-subfams-appr", "lem.fps.prods-mulable-rules.SW1.lem1" ] }, { "id": "ac-details-infinite-products-part-2-part-2", "title": "Details: Infinite products (part 2) (part 2)", "source_path": "AlgebraicCombinatorics/tex/Details/InfiniteProducts2.tex", "target_theorems": [ "lem.fps.subs.rule-infprod-fin" ] }, { "id": "ac-details-domino-tilings", "title": "Details: Domino tilings", "source_path": "AlgebraicCombinatorics/tex/Details/DominoTilings.tex", "target_theorems": [ "def.gf.weighted-set.domino.Rn3.ABC", "prop.gf.weighted-set.domino.Rn3.ABC" ] }, { "id": "ac-details-limits-of-fpss", "title": "Details: Limits of FPSs", "source_path": "AlgebraicCombinatorics/tex/Details/Limits.tex", "target_theorems": [ "prop.fps.lim.sum-prod.K", "prop.fps.lim.sum-prod.L" ] }, { "id": "ac-details-laurent-power-series", "title": "Details: Laurent power series", "source_path": "AlgebraicCombinatorics/tex/Details/LaurentSeries.tex", "target_theorems": [ "lem.fps.laure.xa", "prop.fps.laure.xk" ] } ] }