Teorema mestre de MacMahon

MacMahon Master theorem In mathematics, the MacMahon Master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916). It is often used to derive binomial identities, most notably Dixon's identity.

Conteúdo 1 Fundo 2 Precise statement 3 Derivation of Dixon's identity 4 Veja também 5 References Background In the monograph, MacMahon found so many applications of his result, he called it "a master theorem in the Theory of Permutations." He explained the title as follows: "a Master Theorem from the masterly and rapid fashion in which it deals with various questions otherwise troublesome to solve."

The result was re-derived (with attribution) a number of times, most notably by I. J. Good who derived it from his multilinear generalization of the Lagrange inversion theorem. MMT was also popularized by Carlitz who found an exponential power series version. Dentro 1962, Good found a short proof of Dixon's identity from MMT. Dentro 1969, Cartier and Foata found a new proof of MMT by combining algebraic and bijective ideas (built on Foata's thesis) and further applications to combinatorics on words, introducing the concept of traces. Desde então, MMT has become a standard tool in enumerative combinatorics.

Although various q-Dixon identities have been known for decades, except for a Krattenthaler–Schlosser extension (1999), the proper q-analog of MMT remained elusive. After Garoufalidis–Lê–Zeilberger's quantum extension (2006), a number of noncommutative extensions were developed by Foata–Han, Konvalinka–Pak, and Etingof–Pak. Further connections to Koszul algebra and quasideterminants were also found by Hai–Lorentz, Hai–Kriegk–Lorenz, Konvalinka–Pak, e outros.

Finalmente, according to J. D. Louck, theoretical physicist Julian Schwinger re-discovered the MMT in the context of his generating function approach to the angular momentum theory of many-particle systems. Louck writes: It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents.[1] Precise statement Let {estilo de exibição A=(uma_{eu j})_{mtimes m}} be a complex matrix, e deixar {estilo de exibição x_{1},ldots ,x_{m}} be formal variables. Consider a coefficient {estilo de exibição G(k_{1},pontos ,k_{m}),=,{De repente [}x_{1}^{k_{1}}cdots x_{m}^{k_{m}}{maior ]},prod _{i=1}^{m}{De repente (}uma_{i1}x_{1}+pontos +a_{Eu estou}x_{m}{De repente )}^{k_{eu}}.} (Here the notation {estilo de exibição [f]g} significa "the coefficient of monomial {estilo de exibição f} dentro {estilo de exibição g} ".) Deixar {estilo de exibição t_{1},ldots ,t_{m}} be another set of formal variables, e deixar {displaystyle T=(delta _{eu j}t_{eu})_{mtimes m}} be a diagonal matrix. Então {soma de estilo de exibição _{(k_{1},pontos ,k_{m})}G(k_{1},pontos ,k_{m}),t_{1}^{k_{1}}cdots t_{m}^{k_{m}},=,{fratura {1}{a(EU_{m}-TA)}},} where the sum runs over all nonnegative integer vectors {estilo de exibição (k_{1},pontos ,k_{m})} , e {estilo de exibição I_{m}} denotes the identity matrix of size {estilo de exibição m} .

Derivation of Dixon's identity Consider a matrix {estilo de exibição A={começar{pmatrix}0&1&-1\-1&0&1\1&-1&0end{pmatrix}}.} Compute the coefficients G(2n, 2n, 2n) directly from the definition: {estilo de exibição {começar{alinhado}G(2n,2n,2n)&={De repente [}x_{1}^{2n}x_{2}^{2n}x_{3}^{2n}{De repente ]}(x_{2}-x_{3})^{2n}(x_{3}-x_{1})^{2n}(x_{1}-x_{2})^{2n}\[6pt]&=,soma _{k=0}^{2n}(-1)^{k}{alguns deles {2n}{k}}^{3},fim{alinhado}}} where the last equality follows from the fact that on the right-hand side we have the product of the following coefficients: {estilo de exibição [x_{2}^{k}x_{3}^{2n-k}](x_{2}-x_{3})^{2n}, [x_{3}^{k}x_{1}^{2n-k}](x_{3}-x_{1})^{2n}, [x_{1}^{k}x_{2}^{2n-k}](x_{1}-x_{2})^{2n},} which are computed from the binomial theorem. Por outro lado, we can compute the determinant explicitly: {exibi-lo(I-TA),=,a {começar{pmatrix}1&-t_{1}&t_{1}\t_{2}&1&-t_{2}\-t_{3}&t_{3}&1end{pmatrix}},=,1+{De repente (}t_{1}t_{2}+t_{1}t_{3}+t_{2}t_{3}{maior )}.} Portanto, by the MMT, we have a new formula for the same coefficients: {estilo de exibição {começar{alinhado}G(2n,2n,2n)&={De repente [}t_{1}^{2n}t_{2}^{2n}t_{3}^{2n}{De repente ]}(-1)^{3n}{De repente (}t_{1}t_{2}+t_{1}t_{3}+t_{2}t_{3}{maior )}^{3n}\[6pt]&=(-1)^{n}{alguns deles {3n}{n,n,n}},fim{alinhado}}} where the last equality follows from the fact that we need to use an equal number of times all three terms in the power. Now equating the two formulas for coefficients G(2n, 2n, 2n) we obtain an equivalent version of Dixon's identity: {soma de estilo de exibição _{k=0}^{2n}(-1)^{k}{alguns deles {2n}{k}}^{3}=(-1)^{n}{alguns deles {3n}{n,n,n}}.} See also Permanent References ^ Louck, James D. (2008). Unitary symmetry and combinatorics. Singapore: Mundial Científico. pp. viii. ISBN 978-981-281-472-2. P.A.. MacMahon, Combinatory analysis, vols 1 e 2, Cambridge University Press, 1915-16. Good, I.J. (1962). "A short proof of MacMahon's 'Master Theorem'". Proc. Cambridge Philos. Soc. 58 (1): 160. Bibcode:1962PCPS...58..160G. doi:10.1017/S0305004100036318. Zbl 0108.25104. Good, I.J. (1962). "Proofs of some 'binomial' identities by means of MacMahon's 'Master Theorem'". Proc. Cambridge Philos. Soc. 58 (1): 161–162. Bibcode:1962PCPS...58..161G. doi:10.1017/S030500410003632X. Zbl 0108.25105. P. Cartier and D. foate, Problèmes combinatoires de commutation et réarrangements, Notas de aula em matemática, não. 85, Springer, Berlim, 1969. eu. Carlitz, An Application of MacMahon's Master Theorem, SIAM Journal on Applied Mathematics 26 (1974), 431–436. I.P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley, Nova york, 1983. C. Krattenthaler and M. Schlosser, A new multidimensional matrix inverse with applications to multiple q-series, Discrete Math. 204 (1999), 249-279. S. Garoufalidis, T. T. Q. Lê and D. Zeilberger, The Quantum MacMahon Master Theorem, Proc. Nacional. Acad. of Sci. 103 (2006), não. 38, 13928–13931 (eprint). M. Konvalinka and I. Pak, Non-commutative extensions of the MacMahon Master Theorem, Adv. Matemática. 216 (2007), não. 1. (eprint). D. Foata and G.-N. Han, A new proof of the Garoufalidis-Lê-Zeilberger Quantum MacMahon Master Theorem, J. Álgebra 307 (2007), não. 1, 424–431 (eprint). D. Foata and G.-N. Han, Specializations and extensions of the quantum MacMahon Master Theorem, Linear Algebra Appl 423 (2007), não. 2-3, 445–455 (eprint). P.H. Hai and M. Lorenz, Koszul algebras and the quantum MacMahon master theorem, Touro. Londres. Matemática. Soc. 39 (2007), não. 4, 667–676. (eprint). P. Etingof and I. Pak, An algebraic extension of the MacMahon master theorem, Proc. América. Matemática. Soc. 136 (2008), não. 7, 2279–2288 (eprint). P.H. Hai, B. Kriegk and M. Lorenz, N-homogeneous superalgebras, J. Noncommut. Geometria. 2 (2008) 1–51 (eprint). J.D. Louck, Unitary symmetry and combinatorics, World Sci., Hackensack, Nova Jersey, 2008. Categorias: Enumerative combinatoricsFactorial and binomial topicsTheorems in combinatoricsTheorems in linear algebra