Assume I have a collection of elements $(x_{ij})_{i \in I, j\in J }$. I want to express in a concise notation the sum of the product of the combinations, where one of the elements is left out from each term of the product.
To clarify, I will make the following example with $I = J = \{1,2\}$. I want to write the following expression in a concise form
$$
x_{11}x_{12}x_{21} + x_{11}x_{22}x_{12} + x_{12}x_{21}x_{22}+ x_{11}x_{21}x_{22},
$$
i.e., a sum of products of all combinations over $I$ and $J$ where one of the elements is left out.
How to do it for a generic $I$ and $J$? I tried to write it as a sum of products but I am not very familiar with combinations.
Best Answer
This is called an elementary symmetric polynomial. For your example the notation is $$e_3(x_{11}, x_{12}, x_{21}, x_{22})$$ and I imagine nobody would object if you abbreviated this to $$e_3(x_{11}\ldots x_{22})$$ or $$e_3(x_{ij}).$$