The elementary symmetric polynomial $e_{n,k}$ in variables $x_1,\ldots,x_n$ is the sum of all products of $k$ of those $n$ variables, thus: $$ e_{n,k} = \sum_{\begin{smallmatrix} A \, \subseteq\, \{1,\ldots,n\} \\ |A| \, =\, k \end{smallmatrix}} \prod_{j\,\in\, A} x_j. $$ Now suppose I want to speak of putting the terms in a particular order, here exemplified by $e_{6,3}:$
\begin{align} e_{6,3} = {} \qquad& x_1 x_2 x_3 & & \text{max index} = 3 \\[10pt] {}+{} & \underbrace{\overbrace{x_1 x_2 x_4}^{\text{max}\,=\,2} + x_1 x_3 x_4 + x_2 x_3 x_4}_{\text{max} = 3} & & \text{max index} = 4 \\[10pt] {}+{} & \underbrace{\underbrace{\overbrace{x_1 x_2 x_5}^{\text{max} = 2} + x_1 x_3 x_5 + x_2 x_3 x_5}_{\text{max} = 3} + x_1 x_4 x_5 + x_2 x_4 x_5 + x_3 x_4 x_5}_{\text{max}=4} & & \text{max index} = 5 \\[10pt] {} + {} & \overbrace{\overbrace{\overbrace{x_1 x_2 x_6}^{\text{max}=2} {} + x_1 x_3 x_6 + x_2 x_3 x_6}^{\text{max}=3} {} + x_1 x_4 x_6 + x_2 x_4 x_6 + x_3 x_4 x_6}^{\text{max}=4} \\ & {} + x_1 x_5 x_6 + x_2 x_5 x_6 + x_3 x_5 x_6 + x_4 x_ 5 x_6 & & \text{max index} = 6 \end{align}
First we have the terms in which the maximum index is 3, then 4, then 5 then 6.
Within each of those, we first have the ones with maximum second-largest index 2, then (where applicable) 3, then (where applicable) 4, etc.
Within those, the third-largest index increases step by step.
And so on.
I thought "lexicographic order", and then immediately noticed that that is not what this is; for example, $x_3 x_4 x_5$ comes before $x_1 x_2 x_6$. And notice that if we were ordering the terms of $e_{7,3},$ then all of the above would come immediately before $x_1 x_2 x_7,$ so that $x_4 x_5 x_6$ comes before $x_1 x_2 x_7.$
So my question is: Is there some standard name for this order?
Unlike lexicographic order, this order has the property that if you add another variable (in this case $x_7$), the additional terms all come after the ones you've already got.
