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I'll give a rough answer whose utility might depend on to what extent you want to answer the question in theory vs. practice. The given problem is at least morally reducible to the problem #KNAPSAC …

No. Here is a counterexample. Let $\mathcal{A} = \{1,2,3,4\}$, $\mathcal{B} = \{1,2\}$, and $f(x) = g(x) = \lceil\frac{x}{2}\rceil$. Let the joint probability mass function of $X$ and $Y$ be given …

Here is my original answer (see below for a better one): Writing down \[ g(p) = \sum_{k=0}^n h(k)\binom{n}{k}p^k(1-p)^{n-k} \] and differentiating twice gives \[ g''(p) = \sum_{k=0}^n h(k)\binom{n} …

You can achieve $\lvert F\rvert = 2\lceil\log_2 n\rceil$ by using all subsets of the form $\{x\in X \vert i^{\text{th}}\text{ bit of }x\text{ is }j\}$ for $i \in \{0,1,\ldots,\lceil\log_2 n\rceil-1\}$ …

If I understand you correctly, you have certain sets of points which you would like to count up to permutations of the coordinates. This is the same as counting lists of points up to permutations of …

The standard proofs of Sperner's Lemma are the first thing that comes to my mind in this context. They don't have exactly the form you mentioned; in particular they're not really inductive. Nonethel …