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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 $a_i.$

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This shows up in the literature under the name score sequence. For the $r=2$ case you can find more information on MathWorld and on the page of A000571 in the OEIS.

For general values of $r$ a characterization is given in On Score Sequences of $k$-Hypertournaments 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 $j=m$.

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