# Multiset Notation

There is a multiset $A$, of length $n$ that can contain only $1s$ or $0s$. How would I notate that? How about for a multiset that could contain any number from $1-1000$, or that could contain any real number?

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What exactly is a binary multiset? Googling "binary multiset" turns up nothing relevant. – Rahul Aug 4 '11 at 15:48
@Rahul edited. thanks for the comment. – Matt Munson Aug 4 '11 at 16:02
I'd just stick to plain old English. – anon Aug 4 '11 at 16:08

A multiset of length $n$ that contains only $0$s and $1$s can be described as a multiset of cardinality $n$ whose underlying set is a subset of $\{0,1\}$. If you need to be formal, this is an ordered pair $\langle S,m \rangle$ such that $S \subseteq \{0,1\}$, $m:S \to \mathbb{Z}^+$, and $\sum\limits_{s \in S}m(s) = n$. Here $S$ is the underlying set, and $m$ is the multiplicity function. You can of course replace $\{0,1\}$ by any other set: $\{n \in \mathbb{Z}:1 \le n \le 1000\}$, $\mathbb{R}$, or whatever.
Why not get rid of the $S$ and just keep $m : \{0,1\} \to \mathbb N$, where $\mathbb N$ includes $0$? – Rahul Aug 5 '11 at 2:24
@Rahul: It’s largely a matter of convention: most authors (in my experience) disallow $0$ as a multiplicity. There are some notable exceptions, however: Richard P. Stanley does exactly what you suggest in his Enumerative Combinatorics. – Brian M. Scott Aug 5 '11 at 2:46