I was studying a book (pdf) called "Generating Functionology" and the author gives the definition of cards and hands for the purpose of combinatorics. However I did not understand the definition of hands.
Definition. A card $C(S,p)$ is a pair consisting of a finite set $S$ (the ‘label set’) of positive integers, and a picture $p ∈ P$. The weight of $C$ is $n = |S|$. A card of weight n is called standard if its label set is $[n]$*.
- Recall that $[n]$ is the set ${1, 2,...,n}$.
Definition. A hand $H$ is a set of cards whose label sets form a partition of $[n]$, for some $n$. This means that if $n$ denotes the sum of the weights of the cards in the hand, then the label sets of the cards in $H$ are pairwise disjoint, nonempty, and their union is $[n]$.
Am I right in saying that this means that it is the set of all the cards (from numbers $1$ to $n$) of the same picture $p$? Or does it mean that it could be set of subset of $n$ cards (from numbers $1$ to $n$) not necessarily of same picture?
No, it is not necessary that all cards in a hand have the same picture. Here is a an illustration from the first edition of the book that appears in Example 2 of section 3.4. The second edition does not include this illustration for some reason, although it does include the same example. As you can see, the three cards in the hand have three different pictures.
(The example relates to the set of all permutations as "an exponential family" in Wilf's terminology. The goal is to develop generating functions that yield information about the various kinds of permutations given numbers and sizes of cycles.)