# Cauchy's one-line notation for sum over ordered permutations.

Setup: I would like to use Cauchy's 1-line notation for permutations to express the following kind of sum more compactly:

\begin{align} C_{1,2} C_{3,4}+C_{1,3}C_{2,4}+C_{1,4}C_{2,3}, \tag{*}\label{*} \end{align} where $C_{i,j}$ are some indexed coefficients. i.e., all the perumutations of $C_{i,j}C_{k,l}$ for $i,j,k,l=1,2,3,4$ with $i<j$, $k<l$, and $i<k$.

Example: If my sum had instead been over products like $C_{i,j}D_{k,l}$ with permutations of $i,j,k,l=1,2,3,4$ with no ordering restriction, then I could compactly write,

\begin{align} \sum_\sigma C_{\sigma(1),\sigma(2)}D_{\sigma(3),\sigma(4)}, \end{align} where $\sigma$ denotes the permutations of $1,2,3,4$.

Question: To generate \eqref{*}, I need to take into account the fact that $C_{i,j}C_{k,l}=C_{k,l}C_{i,j}$ and implement the ordering restriction. Is there a compact way to achieve this using Cauchy's 1-line notation (and perhaps other "standard" combinatoric notation/conventions)?

• Could you maybe explain what $C_{i,j}$ is exactly? – Benji Altman Dec 17 '17 at 1:37
• Perhaps they could be Plucker coordinates in a quadratic relation – Somos Dec 17 '17 at 2:56