# Set notation: subtracting elements with given cardinality from the powerset

I have a set $S = \{1,2,\ldots,n\}$ of $n$ elements and I denote with $P(S)$ the powerset of $S$.

Which is a correct and accepted notation to say that the set $Z$ is composed by all the elements in $P(S)$ with the exception of all the subsets of cardinality $h$?

Example: if $S=\{1,2,3\}$ then $P(S) = \{\emptyset, \{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$.

If $h = 1$, then it should be $Z = \{\emptyset, \{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}=P(S)\setminus\{\{1\},\{2\},\{3\}\}$.

How can it be expressed in a formal and concise way, for any value of $n$ and $h$?

Thanks

• The notations $[X]^h$ and $X^{[h]}$ are both used to denote the collection of subsets of $X$ of size $h$, so one possibility is $\mathcal P(S)\setminus[S]^h$. I haven't seen any specific notation for this set, though $[S]^{\ne h}$ would be reasonable. Commented Aug 6, 2012 at 23:04
• I've also seen $\binom{X}{h}$ for $h$-subsets of $X$, but I can't find a reference (so it's obviously not standard). Another option is to take this further and write $\mathcal{P}_{\ne h}(S)$. (Also note $[n]=\{1,\cdots,n\}$ is fairly standard shorthand in analytic number theory and combinatorics.)
– anon
Commented Aug 6, 2012 at 23:06

$$Z = \left \{ x | x \in P(S) , |x| \neq h \right\}$$
$$Z = \{x \in P(S) : |x| \neq h\}$$