Let's say I have a list of sets $S_i$, for $i=1,\ldots,n$. We often write the cartesian product of all these sets, with the exception of $S_k$ as:
$$S=S_1\times\cdots\times S_{k-1}\times S_{k+1}\times\cdots\times S_n$$
Is there a more succinct way to write it?
In Wikipedia (https://en.wikipedia.org/wiki/Cartesian_product), I found something, which might be what you are looking for: $\prod_{n=1}^k \Bbb{R} = \Bbb{R}\times \Bbb{R} \times\cdots\times \Bbb{R} = \Bbb{R}^k$. So maybe something like this one is also valid: $$\prod_{\scriptstyle i = 1\atop\scriptstyle i \ne k}^nS_i$$
where $S_i$ is the $i^\text{th}$ set of the list you mentioned.
\prod_{i = 1\atop i \ne k}^n would be better? (with \atop separating lines instead of a comma) $\endgroup$ \prod_{\scriptstyle i = 1\atop\scriptstyle i \ne k}^n: compare$$\prod_{i = 1\atop i \ne k}^nS_i\quad\hbox{and}\quad \prod_{\scriptstyle i = 1\atop\scriptstyle i \ne k}^nS_i$$ $\endgroup$ I have seen a notation for this kind of construction during some of my math lectures (but can't find a reference right now). This was mostly in the context of differential forms (e.g. interior product with vector), but can be applied to your case: $$ S_1\times \dotsm \times \widehat{S_k} \times\dotsm \times S_n := S_1\times \dotsm \times S_{k-1}\times S_{k+1} \times \dotsm S_n$$ The hat denotes the factor to be omitted. Note that this is not a universally standard notation, so even the professors that used it defined it at some point early in the lecture.
In general, we can write
$$S_1 \times \dots \times S_n := \prod_{i=1}^n S_i$$
and then we can apply all conventions we are used to.
As for your question, this can be written as:
$$\prod_{i = 1 \atop i \neq k}^n S_i$$
Although it might not be common in set theory, it is common for game theorists to write $S_{-i}$ for $S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n$. See page 15 of chapter one of Osborne and Rubinstein's text on game theory, for example.
That notation is useful in game theory because, if $S_j$ represents the set of strategies available to player $j$, then one often needs to describe how all players except player $i$ have acted. Such a description will be a member of $S_i$. In particular, the notation becomes useful in defining a Nash equilibrium. See https://en.wikipedia.org/wiki/Nash_equilibrium.
