Result in enumerative combinatorics and linear algebra
In mathematics, MacMahon's 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.
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. In 1962, Good found a short proof of Dixon's identity from MMT. In 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. Since then, 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, and others.
Finally, according to J. D. Louck, the 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:
 ;templatestyles src="Template:Blockquote/styles.css" / ;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.
Precise statement.
Let formula_1 be a complex matrix, and let formula_2 be formal variables. Consider a coefficient
formula_3
(Here the notation formula_4 means "the coefficient of monomial formula_5 in formula_6".) Let formula_7 be another set of formal variables, and let formula_8 be a diagonal matrix. Then
formula_9
where the sum runs over all nonnegative integer vectors formula_10,
and formula_11 denotes the identity matrix of size formula_12.
Derivation of Dixon's identity.
Consider a matrix
formula_13
Compute the coefficients "G"(2"n", 2"n", 2"n") directly from the definition:
formula_14
where the last equality follows from the fact that on the right-hand side we have the product of the following coefficients:
formula_15
which are computed from the binomial theorem. On the other hand, we can compute the determinant explicitly:
formula_16
Therefore, by the MMT, we have a new formula for the same coefficients:
 formula_17
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"(2"n", 2"n", 2"n") we obtain an equivalent version of Dixon's identity:
formula_18
References.
 ;templatestyles src="Reflist/styles.css" / ;