I am reading Generating Functionology by Herbert S. Wilf in around page -75, he defines the following objects:
Definition. A card C(S, p) is a pair consisting of a finite set S (the ‘label set’) of positive integers, and a picture $p \in P$. The weight of C is $n = |S|$. A card of weight n is called standard if its label set is $[n]$.*
Definition. The weight of a hand is the sum of the weights of the cards in the hand.
Definition. A relabeling of a card C(S, p) with a set S0 is defined if $|S| =|S '|$, and it is the card $C(S', p)$. If $S0 = [|S|] $then we have the standard relabeling of the card.
Definition. A deck $D $is a finite set of standard cards whose weights are all the same and whose pictures are all different. The weight of the deck is the common weight of all of the cards in the deck.
Definition. An exponential family F is a collection of decks $D_1, D_2, . . .$ where for each $n = 1, 2, . . ., $the deck $D_n$ is of weight $n.$
These were a lot of things defined at once, and I feel it quite hard to digest them. I seek an answer which gives a mental model for thinking about the objects defined, like a simple intuitive view of all of them.
I think the particular point troubling me after reading this stack is about the concept of label sets
Introduction
The most fundamental connection between the fields of analysis and combinatorics is the notion of a generating function. The fact that counts of certain combinatorial sets can become coefficients of a series is important , because on one side lies the discreteness of set counting techniques, and on the other side lies the continuous notions of differentiation/integration, and we can swiftly move between these notions by considering the generating function.
The concepts presented, of a card, hand, deck, and an exponential family are meant to capture an entire counting problem.
A counting problem is the problem of counting the number of elements of a set, given a description of that set in some way. For example, consider the set of subsets of $[n]$ of size $k\leq n$. A counting problem would be : find this number for all $n$ and $k \leq n$. The answer is $\frac{n!}{(n-k)!k!}$ for a given $n$ and $k$, but to give the answer for every $n,k$ all at once I can just give you a generating function, and you just find the coefficient of some term depending on $n$ and $k$, you will land up with the answer for that $n$ and $k$. Of course, estimates, asymptotics and other properties of the count sequence will be easy to analyze once you have the generating function.
However, every counting problem boils down to counting the sets consisting of building blocks. This is because, if you can count the building block sets, then every other set can be built out of them, and then using unions, intersections, cartesian products etc. you can easily find the size of another set.
Now, roughly this is the break-up :
Card : A fundamental building block.
Hand : A set built out of cards.
Deck : A set of building blocks, in which each element shares something in common with the others.
Exponential family : The entire set of building blocks, given the entire set of decks.
Card
A card is essentially a set and a picture. Now what is this picture, and what is this label? Essentially, the label is a subset of positive numbers, so you can think of the label as being some set of numbers to associate to the card.
What about the picture? The picture is just a way to associate the labels together to create the ultimate object we want to count. For example, if we want to count graphs, then a label set would label the vertices, but how they are connected can only be specified by a picture. Similarly, if we were counting words, then a label set would tells us what letters are used, but what order they come in is down to a picture.
So, the object that is represented by a card, is a set of numbers and then how the object is created by patching that set of numbers together. The word picture is deliberately used in an abstract sense, because the object we want to count could be very abstract itself! (The number of ways I wish to dress my doll up for a kitty party with the dormouse)
Now, it turns out that a card should be fundamental, much like an element is to chemistry : it should not be "breakable" into simpler mathematical objects that aren't cards already.
The card is then meant to be the building block for the objects that we ultimately wish to count : essentially, our objects of interest are built out of cards. Much like a hand is made of playing cards, which brings us to
Hand
A hand is the mathematical object we want to count. It is created by taking different appropriately cards together, so that the pictures combine and give us a picture of the hand.
The reason for calling it a hand is from the fact that in card games, a set of cards is kept in ones hand during the game, so we call it a hand.
Now, the starting point of solving a counting problem is expressing every object you want to count, as a hand. For example, for counting permutations, you know that a permutation can be written as a product of disjoint cycles (which commute with each other, so there's no order involved in the product), therefore a permutation is hand where each card is a cyclic permutation. Similarly, every finite graph is a disjoint graph-union of connected graphs, so a finite graph is a hand with each card being a connected graph.
So we are counting hands : a hand is made by a mix-and-match of various cards of different types. So it is only fair that we combine the cards of various types together. Which brings us to a
Deck
A deck is a collection of cards that share the same type.
What is the fundamental reason for putting these together? It is because sizes of sub-collections share an obvious relation to the collection which they create by free-choice : by multiplication.
To give an example, say I'm dressing my doll for a party! I have four hair styles to choose from (though I prefer letting hers loose, but that's out because it's not allowed in this party), three skirts (red,red and red in color, but different shades of red. Also every skirt comes with a top as well!) and two shoes (both heeled, she wants to impress her doll friend). Knowing that her ultimate look is a mix-and-match of a hair style, a skirt and a shoe, it is important that I sort the shoes, skirts and hair styles (abstractly) into different piles and count them differently so I can find the number of ways in which I can dress my doll.
So counting the fundamentals , with a mix-and-match means I'm counting what I want eventually. But sorting the fundamentals is important, which is what a deck is doing. A set of decks together creates the entire setting of the problem ,the
Exponential family
An exponential family is a family of decks, so basically every single deck available to you. Note that this completely describes all the fundamental building blocks of the objects you want to count.
For example, knowing how to dress my doll means I need to know everything about what I have to dress her up with : that's the exponential family for that problem.
Thus, at the outset of any counting problem is the exponential family, the notions are near inseparable.
How it all fits together
So to solve a counting problem (from scratch):
Think about what a card is for the problem : the ultimate fundamental object which combines and creates every kind of object you want to count. This really depends on the situation : for example, for permutations we have cycles, but for bipartite graphs, a card is actually a $2$-colored connected labeled bipartite graph (explained on page $88$).
Put the cards into decks : this is fairly natural, in the sense that cards from the same deck can't be mixed.
Creating the exponential family by putting the decks you need together. Sometimes you won't need certain decks : for example, if you choose to count permutations with some property which don't have length bigger than $30$, you can discard decks whose weight is bigger than $30$.
Now, you can count what you want, the setup is made.
Reasons for introducing these terms
There are a variety of reasons why we choose to begin with a high amount of generality. Some of these are :
The fact that we can choose to move to a specific problem, by making it a sub-instance of a larger problem, is a great advantage because we keep the framework while changing the problem. For example, counting permutations and counting permutations of size less than $30$ both come under this framework, where for the second case you can throw out all decks of size more than $30$.
The fact that within the framework, it is not only the hands we can count now. We can count the deck sizes if we know the size of the hands as well! (Using an inversion formula)
Last but not the least, the framework is flexible in the sense that a hand is picked to be a decomposition of $[n]$, but there might be an order to picking the cards as well , so next you could define a hand by not just a set of cards but rather an ordered list of cards. Then the notion of counting changes, but the framework changes ever so slightly to accommodate it. Beginning in generality, means you don't have to stretch to accommodate even far-off looking counting problems.
Short note on this last point : a hand in the given case is formed by putting together a set of cards. If the order of the cards mattered, we would not study $\mathcal{H}(x,y)$ by that expression, but rather by a different expression. The point is, the framework still accommodates all of this.