I'm trying to implement this paper to index non-equivalent poker hands. I have all of the indexing implemented for a given suit configuration. But how can I calculate the size of a suit configuration to determine it's offset in the index?
I beleive this question may be related to What is the Group of symmetries of Texas Holdem Poker hands?
As a simple example, take a deal of 2 cards from a standard deck. There are 169 non equivalent hands. These hands consist of 2 suit configurations:
- [2]: Both cards are the same suit
- [1][1]: Cards are from different suits
The size of [2] is $\binom{13}{2}=78$. And the size of [1][1] is 13 choose 2 with replacement $=91$. These configurations together represent the 169 non-equivalent hands.
However, the approach for calculating [1][1] only works when there is at most a single card drawn from each suit -- it can be modeled as combinations with replacement.
How could I calculate the size of an arbitrary suit configuration? e.g. [2][1], 2 cards of 1 suit, 1 of another suit. Or [2][2][1].
Lets write $[x_1, y_1][x_2, y_2]\ldots$ for number configuration with $y_1$ suits of $x_1$ cards, $y_2$ suits of $x_2$ cards and so on. We have $[x_1, y_1]\ldots = \prod_i [x_i, y_i]$ - suits with different number of cards are indepdent.
Now, for single suit of $x$ cards, we have $13 \choose x$ variants. We now need to choose without ordering and with replacement one variant for each of this variant, and this number is known as multiset coefficient. Taking formula for it, we get $[x, y] = {{{13 \choose{x}} + y - 1}\choose y}$.