Given an exponential family $\mathbb{F}$. For each $n \geq 0 $, and $ k \geq 1$, let $h(n,k)$ denote the numebr of hands $H$ of weight $n$ that consist of $k$ cards, and are such that each card in the hand is a relabeling of some card in some deck in $\mathbb{F}$. Repetitions are allowed, that is we are permitted to take several copies of the same card from one deck and to relabel those copies with different label sets.
How can we expression $h(n,k)$ in terms of $d_1,d_2..,,$ where $d_i$ is the number of different cards in deck $D_i$ ($ i\geq 1)$
I'm trying to make sure I understood this definition correctly, suppose I have an exponential family with one deck:
Then, if I were to calculate $h(2,2)$ would the answer be $4$? I think these would be the selections:
I had much earlier asked a question on Hand, Deck and expt family here
