Let $A=\{a\_1,\ldots,a\_m\}.$ Let a choice function $f:\mathcal{P}(A) \mapsto A,$ be such that, for all $B \in \mathcal{P}(A),$ $f(B)=x$ for some $x\in B.$ For instance, with $A=\{a\_1,a\_2,a\_3\}$ one such function is $f(\{a\_1,a\_2,a\_3\})=a\_1, f(\{a\_1,a\_2\})=a\_1, f(\{a\_1,a\_3\})=a\_1, f(\{a\_2,a\_3\})=a\_2.$ [$\mathcal{P}(A)$ is the set of all non-empty subsets of $A.$]


Let us restrict our attention to all $B\subseteq A$ of cardinality $r,$ i.e., $|B|=r.$ Let $x\_i=|\{B|B \subseteq A, |B|=r, f(B)=a\_i\}|.$ And suppose $x\_{(1)},x\_{(2)},\ldots, x\_{(m)}$ are these frequencies in non-decreasing order. Let us call this vector $x^{m,r,f}=(x\_{(m)},x\_{(m-1)},\ldots, x\_{(1)})$ and $x^{m,r}=\{x^{m,r,f}|f \text{ is a choice function }\}.$


Thus, $x^{3,2}=\{(2,1,0),(1,1,1)\},$ and $x^{4,2}=\{(3,2,1,0),(2,2,2,0),(2,2,1,1),(3,1,1,1)\}.$ 


My question is: Is there any general characterization of the vectors in $x^{m,r}?$ Or at least for $r=2?$ Some partial results will be also be very helpful. 


Note: For $r=2,$ let $\Gamma=(E,A)$ be the directed graph where $A$ is the set of nodes. $(a\_i,a\_j)\in E \iff f(\{a\_i,a\_j\})=a\_i.$ Therefore, $x\_i$ can be interpreted as out-degree of the node $a\_i.$



Configuration of vectors satisfying some constraints
This shows up in the literature under the name *score sequence*. For the $r=2$ case you can find more information on [MathWorld](http://mathworld.wolfram.com/ScoreSequence.html) and on the page of  [A000571](http://oeis.org/A000571) in the OEIS.


For general values of $r$ a characterization is given in [On Score Sequences of $k$-Hypertournaments](https://doi.org/10.1006/eujc.2000.0393) by Zhou Guofei, Yao Tianxing, and Zhang Kemin. In the notation of the post we have that $(x\_{(1)}, \dots, x\_{(m)}) \in x^{m,r}$ if and only if
$$ \sum\_{i = 1}^j x\_{(i)} \geq \binom{j}{r}$$
for all $j$ with equality when $j=m$.


