Hi everyone!

Mandatory orz to Elegia whose blog introduced me to MMT as an elegant approach to prove Dixon's identity.

Today, I would like to write about MacMahon's master theorem (MMT). It is a nice result on the intersection of combinatorics and linear algebra that provides an easy proof to some particularly obscure combinatorial identities, such as the **Dixon's identity**:

Besides that, MMT has large implications in Quantum Physics, which we will hopefully discuss. For now, let's formulate MMT.

**MMT**. Let $$$\mathbf A = [a_{ij}]_{n\times n}$$$, $$$\mathbf X = \operatorname{diag}(x_1,\dots,x_n)$$$, $$$\mathbf t = (t_1,\dots,t_n)$$$, $$$\mathbf x = (x_1,\dots,x_n)$$$ and $$$\mathbf k = (k_1,\dots,k_n) \geq 0$$$. Then,

where $$$\mathbf t^\mathbf k$$$ stands for $$$t_1^{k_1} \dots t_n^{k_n}$$$, and $$$\mathbf x^\mathbf k$$$, correspondingly, for $$$x_1^{k_1} \dots x_n^{k_n}$$$, and $$$\mathbf I = [\delta_{ij}]_{n\times n}$$$ is the identity matrix.