MacMahon Master theorem

For a basic result in analysis of algorithms, see Master theorem.

The MacMahon Master theorem (MMT) is a result in enumerative combinatorics and linear algebra, both branches of mathematics. 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.

Contents

Background

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 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. Since then, MMT became 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, 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.

Precise statement

Let A = (a_{ij})_{m\times m} be a complex matrix, and let x_1,\ldots,x_m be formal variables. Consider a coefficient

G(k_1,\dots,k_m) \, = \, \bigl[x_1^{k_1}\cdots x_m^{k_m}\bigr] \, \prod_{i=1}^m \bigl(a_{i1}x_1 %2B \dots %2B a_{im}x_m \bigl)^{k_i}.

Let t_1,\ldots,t_m be another set of formal variables, and let T = (\delta_{ij}t_i)_{m\times m} be a diagonal matrix. Then

\sum_{(k_1,\dots,k_m)} G(k_1,\dots,k_m) \, t_1^{k_1}\cdots t_m^{k_m} \, = \, \frac{1}{\det (I_m - TA)},

where the sum runs over all nonnegative integer vectors (k_1,\dots,k_m), and I_m denotes the identity matrix of size m.

Derivation of Dixon's identity

Consider a matrix

A = \begin{pmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{pmatrix}.

Compute the coefficients G(2n, 2n, 2n) directly from the definition:

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,

where the last equality follows from the fact that on the r.h.s. we have the product of the following coefficients:

[x_2^k x_3^{2n-k}](x_2 - x_3)^{2n}, \ \ [x_3^k x_1^{2n-k}](x_2 - x_3)^{2n}, \ \ [x_1^k x_2^{2n-k}](x_1 - x_2)^{2n},

which are computed from the binomial theorem. On the other hand, we can compute the determinant explicitly:

\det(I - TA) \, = \, \det \begin{pmatrix} 1 & -t_1 & t_1 \\ t_2 & 1 & -t_2 \\ -t_3 & t_3 & 1 \end{pmatrix} \, = \, 1 %2B \bigl(t_1 t_2 %2B t_1 t_3 %2Bt_2t_3\bigr).

Therefore, by the MMT, we have new formula for the same coefficients:

G(2n,2n,2n) \, = \, \bigl[t_1^{2n}t_2^{2n}t_3^{2n}\bigl] (-1)^{3n} \bigl(t_1 t_2 %2B t_1 t_3 %2Bt_2t_3\bigr)^{3n} \, = \, (-1)^{n} \binom{3n}{n,n,n},

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(2n, 2n, 2n) we obtain an equivalent version of Dixon's identity:

\sum_{k=0}^{2n} (-1)^k \binom{2n}{k}^3\, = \, (-1)^{n} \binom{3n}{n,n,n}.

References