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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?

Real numbers
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  • Are you familiar with the usage of the words "card" and "hand" in popular card games? This should match the intuitive definition, just having made it more formal. – JMoravitz Oct 14 '20 at 12:04
  • @JMoravitz from what I know of hand, all the cards in a hand should have the same picture, but the definition does not say so I think, that's why I'm having a bit of my confusion. – Real numbers Oct 14 '20 at 12:10
  • I think we would need to see what kind of questions are asked about "hands" and how they are solved. It appears we can have cards of different weights (what for?) and cards that have the same weight but different label sets (what for?). A hand could have fewer than $n$ cards because some of the cards may have label sets with more than one element, but it could have multiple distinct pictures (possibly as many distinct pictures as cards). Frankly I am having a very hard time seeing how this relates to any questions involving ordinary playing cards. – David K Oct 14 '20 at 12:57
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    @JMoravitz The concept of "card" and "hand" invented by Wilf are an attempt to define a general class of combinatorial objects to which the theorem he calls "The Exponential Formula" applies. They are not an attempt to axiomatize playing cards. – awkward Oct 14 '20 at 13:14

1 Answers1

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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.)

enter image description here

awkward
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  • It seems the picture is just a graphical representation of a cycle of length $|S|$ with the nodes labeled $1,2,\ldots,|S|$ (independently of the specific elements of the label set; for example the middle card has nodes $1$ and $2$ in the picture but the label set has elements $2$ and $7$). It seems redundant; if this is indeed how it works, we can deduce the picture from the label set. – David K Oct 14 '20 at 17:42
  • @DavidK In this application the pictures are cycles, but in other applications they are other objects, so in other applications of "card" it may not be so simple to deduce the picture from the labels. If you want to explore how Wilf uses "card", "hand" and "deck" in a variety of applications, you can take a look at generatingfunctionology. It's available for free download in pdf format. – awkward Oct 14 '20 at 18:26