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Given a multiset, e. g. S = {1, 1, 2, 3, 4, 4, 5}, what would be a short, concise notation to express the number of unique items in the multiset? (five in the given example).

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The set of unique items in multiset $S$ is usually called the support of $S$, and denoted $\operatorname{supp}S$. So, you can combine this with your preferred notation for number of elements: $\#$ or $|\cdot|$ to form $$\# (\operatorname{supp}S) \qquad \text{or}\qquad |\operatorname{supp}S|$$ Both are reasonably concise and are pretty descriptive.

According to OEIS Wiki, the cardinality of support of a multiset is called its dimension. Following this terminology, one could use $\dim S$. However, I don't find this as intuitive as either of the above options. And the usage of the term dimension may be somewhat narrow.

  • In Cybernetics (which is not mathematics) the magnitude of the support of a multiset is called its "variety". I would agree that dimension is not a good name, but "magnitude of the support of a multiset" sounds rather clunky. Variety seems better because it reduces to the size of a set when no elements are repeated. – pdini Jun 04 '20 at 11:11