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| {{for|a basic result in [[analysis of algorithms]]|Master theorem}}
| | Hi! <br>My name is Estella and I'm a 23 years old boy from Iceland.<br><br>Here is my web blog [http://brucegmusic.com red bottom shoes] |
| The '''MacMahon Master theorem''' ('''MMT''') is a result in [[enumerative combinatorics]] and [[linear algebra]], both branches of [[mathematics]]. It was discovered by [[Percy Alexander MacMahon|Percy MacMahon]] and proved in his monograph ''Combinatory analysis'' (1916). It is often used to derive binomial identities, most notably [[Dixon's identity]].
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| == Background ==
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| In the monograph, MacMahon found so many applications of his result, he called it "a master theorem in the Theory of Permutations." The result was re-derived (with attribution) a number of times, most notably by [[I. J. Good]] who derived it from his mulilinear generalization of the [[Lagrange inversion theorem]]. MMT was also popularized by [[Leonard Carlitz|Carlitz]] who found an [[Exponential generating function#Exponential generating function|exponential]] [[power series]] version. In 1962, Good found a short proof of Dixon's identity from MMT. In 1969, [[Pierre Cartier (mathematician)|Cartier]] and Foata found a new proof of MMT by combining [[algebra]]ic and [[bijective proof|bijective]] ideas (built on Foata's thesis) and further applications to [[combinatorics on words]], introducing the concept of [[trace monoid|trace]]s. Since then, MMT has become a standard tool in enumerative combinatorics.
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| 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 algebra|quantum]] extension (2006), a number of [[Noncommutative geometry|noncommutative]] extensions were developed by Foata–Han, Konvalinka–Pak, and Etingof–Pak. Further connections to [[Koszul algebra]] and [[quasideterminant]]s were also found by Hai–Lorentz, Hai–Kriegk–Lorenz, Konvalinka–Pak, and others.
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| Finally, 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 system]]s. Louck writes:
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| {{quote|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.{{cite quote}}}}
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| == Precise statement ==
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| Let <math>A = (a_{ij})_{m\times m}</math> be a complex matrix, and let <math>x_1,\ldots,x_m</math> be formal variables. Consider a [[coefficient]]
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| :<math>
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| G(k_1,\dots,k_m) \, = \, \bigl[x_1^{k_1}\cdots x_m^{k_m}\bigr] \,
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| \prod_{i=1}^m \bigl(a_{i1}x_1 + \dots + a_{im}x_m \bigl)^{k_i}.
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| </math>
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| Let <math>t_1,\ldots,t_m</math> be another set of formal variables, and let <math>T = (\delta_{ij}t_i)_{m\times m}</math> be a [[diagonal matrix]]. Then
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| :<math>
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| \sum_{(k_1,\dots,k_m)} G(k_1,\dots,k_m) \, t_1^{k_1}\cdots t_m^{k_m} \, = \,
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| \frac{1}{\det (I_m - TA)},
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| </math>
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| where the sum runs over all nonnegative integer vectors <math>(k_1,\dots,k_m)</math>,
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| and <math>I_m</math> denotes the [[identity matrix]] of size <math>m</math>.
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| == Derivation of Dixon's identity ==
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| Consider a matrix
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| :<math>
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| A = \begin{pmatrix}
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| 0 & 1 & -1 \\
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| -1 & 0 & 1 \\
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| 1 & -1 & 0
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| \end{pmatrix}.
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| </math>
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| Compute the coefficients ''G''(2''n'', 2''n'', 2''n'') directly from the definition:
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| :<math>
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| G(2n,2n,2n) = \bigl[x_1^{2n}x_2^{2n}x_3^{2n}\bigl] (x_2 - x_3)^{2n} (x_3 - x_1)^{2n} (x_1 - x_2)^{2n} \, = \, \sum_{k=0}^{2n} (-1)^k \binom{2n}{k}^3,
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| </math>
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| where the last equality follows from the fact that on the r.h.s. we have the product of the following coefficients:
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| :<math>[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},</math>
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| which are computed from the [[binomial theorem]]. On the other hand, we can compute the [[determinant]] explicitly:
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| :<math>
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| \det(I - TA) \, = \, \det \begin{pmatrix}
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| 1 & -t_1 & t_1 \\
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| t_2 & 1 & -t_2 \\
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| -t_3 & t_3 & 1
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| \end{pmatrix} \, = \, 1 + \bigl(t_1 t_2 + t_1 t_3 +t_2t_3\bigr).
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| </math> | |
| Therefore, by the MMT, we have new formula for the same coefficients:
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| :<math>
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| G(2n,2n,2n) \, = \, \bigl[t_1^{2n}t_2^{2n}t_3^{2n}\bigl] (-1)^{3n} \bigl(t_1 t_2 + t_1 t_3 +t_2t_3\bigr)^{3n} \, = \, (-1)^{n} \binom{3n}{n,n,n},
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| </math>
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| where the last equality follows from the fact that we need use an equal number of times the all three terms in the power. Now equating two formulas for coefficients ''G''(2''n'', 2''n'', 2''n'') we obtain an equivalent version of Dixon's identity:
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| :<math> \sum_{k=0}^{2n} (-1)^k \binom{2n}{k}^3\, = \, (-1)^{n} \binom{3n}{n,n,n}.
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| </math>
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| == References ==
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| * P.A. MacMahon, ''[http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009 Combinatory analysis]'', vols 1 and 2, Cambridge University Press, 1915–16.
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| * {{cite journal | zbl=0108.25104 | authorlink=I. J. Good | first=I.J. | last=Good | title=A short proof of MacMahon's ‘Master Theorem’ | journal=[[Proc. Cambridge Philos. Soc.]] | volume=58 | year=1962 | page=160 }}
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| * {{cite journal | zbl=0108.25105 | authorlink=I. J. Good | first=I.J. | last=Good | title=Proofs of some `binomial' identities by means of MacMahon's ‘Master Theorem’ | journal=[[Proc. Cambridge Philos. Soc.]] | volume=58 | year=1962 | pages=161–162 }}
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| * [[Pierre Cartier (mathematician)|P. Cartier]] and D. Foata, [http://www.mat.univie.ac.at/~slc/books/cartfoa.html Problèmes combinatoires de commutation et réarrangements], ''Lecture Notes in Mathematics'', no. 85, Springer, Berlin, 1969.
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| * [[Leonard Carlitz|L. Carlitz]], An Application of MacMahon's Master Theorem, ''SIAM Journal on Applied Mathematics'' 26 (1974), 431–436.
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| * I.P. Goulden and [[David M. Jackson|D. M. Jackson]], ''Combinatorial Enumeration'', John Wiley, New York, 1983.
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| * C. Krattenthaler and M. Schlosser, [http://radon.mat.univie.ac.at/users/kratt/public_html/artikel/minv.ps.gz A new multidimensional matrix inverse with applications to multiple ''q''-series], ''Disc. Math.'' 204 (1999), 249–279.
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| * S. Garoufalidis, T. T. Q. Lê and [[Doron Zeilberger|D. Zeilberger]], [http://www.pnas.org/content/103/38/13928.full The Quantum MacMahon Master Theorem], ''Proc. Natl. Acad. of Sci.'' 103 (2006), no. 38, 13928–13931 ([http://arxiv.org/abs/math/0303319 eprint]).
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| * M. Konvalinka and [[Igor Pak|I. Pak]], Non-commutative extensions of the MacMahon Master Theorem, ''Adv. Math.'' 216 (2007), no. 1. ([http://arxiv.org/abs/math/0607737 eprint]).
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| * D. Foata and G.-N. Han, A new proof of the Garoufalidis-Lê-Zeilberger Quantum MacMahon Master Theorem, ''J. Algebra'' 307 (2007), no. 1, 424–431 ([http://arxiv.org/abs/math/0603464 eprint]).
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| * D. Foata and G.-N. Han, Specializations and extensions of the quantum MacMahon Master Theorem, ''Linear Algebra Appl'' 423 (2007), no. 2–3, 445–455 ([http://arxiv.org/abs/math.CO/0603466 eprint]).
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| * P.H. Hai and M. Lorenz, Koszul algebras and the quantum MacMahon master theorem, ''Bull. Lond. Math. Soc.'' 39 (2007), no. 4, 667–676. ([http://arxiv.org/abs/math/0603169 eprint]).
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| * P. Etingof and I. Pak, An algebraic extension of the MacMahon master theorem, ''Proc. Amer. Math. Soc.'' 136 (2008), no. 7, 2279–2288 ([http://arxiv.org/abs/math/0608005 eprint]).
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| * P.H. Hai, B. Kriegk and M. Lorenz, ''N''-homogeneous superalgebras, ''J. Noncommut. Geom.'' 2 (2008) 1–51 ([http://arxiv.org/abs/0704.1888 eprint]).
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| * J.D. Louck, ''Unitary symmetry and combinatorics'', World Sci., Hackensack, NJ, 2008.
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| [[Category:Enumerative combinatorics]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in combinatorics]]
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| [[Category:Theorems in linear algebra]]
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