Notation for set containing combinations of sets given by multinomial coefficient number of options.

Question

Assume there are sets $A_1,A_2,\dots,A_n$. Let $m\leq n$ and now partition $\{1,\dots,n\}$ into $m$ subsets $N_1,\dots,N_m$. So, there are basically $$\sum_{l_1+\dots+l_m = n} {{n}\choose{l_1,\dots,l_m}}$$ different ways to do this. Let $$B_i = \bigcap_{j\in N_i} A_j \cap \bigcap_{j\in\{1,\dots,n\}\setminus N_i} A_j^c.$$ I am interested in the set containing $\{B_1,\dots,B_m\}$ for all possible partitions of $\{1,\dots,n\}$ into $m$ sets. So something like $$C=\{\{B_1,\dots,B_m\}\ |\ N_1,\dots,N_m \text{ partition of }\{1,\dots,n\} \text{ and } B_i \text{ chosen as above}\}.$$ Is there are more elegant way to define this $C$?

Answer 1

We commonly use $\binom{n}{m}$ to denote the number of $m$-element subsets of a set $S$ with size $n$.

• We sometimes find (e.g. by R. P. Stanley) \begin{align*} \binom{S}{m}:=\{A\subset S\,\big|\, |A|=m\}\tag{1} \end{align*} to denote the set of subsets of $S$ with size $m$.

Inspired by this we recall that ${n\brace m}$, the Stirling numbers of the second kind denote the number of ways to partition a set $S$ of size $n$ into $m$ non-empty subsets.

• Analogously to (1) we could define \begin{align*} \color{blue}{{S\brace m}}:=\{A \textrm{ partition of } \mathcal{P} (S)\,\big|\, |A|=m\} \end{align*} as the set of all partitions $A$ of $S$ with $|A|=m$.