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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

  • 0
    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](http://en.wikipedia.org/wiki/Power_set#Subsets_of_limited_cardinality) and write $\mathcal{P}_{\ne h}(S)$. (Also note $[n]=\{1,\cdots,n\}$ is fairly standard shorthand in analytic number theory and combinatorics.)2012-08-06

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

An alternate, more compact version:

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