Let $(a\_1,a\_2,\dots, a\_n)$ be a sequence of non-negative integers. 


**Q.** When does there exists a simple graph $G$ such that its number of $k$-cliques is $a\_k$ (that is $G$ has $a\_1$ vertices, $a\_2$ edges, $a\_3$ trinagles, etc)? 


Probably, it is hopeless to get a complete description of such sequences, so I am interested in any necessary or sufficient conditions on $\{a\_i\}$.


**Upd.** As Steve Huntsman mentions in the comments there are trivial necessary conditions: $a\_k\le \binom{a\_0}{k}$. There are also more complicated inequalities like 
$$
\binom{a\_1}{3}-a\_3\le \bigl(\binom{a\_1}{2}-a\_2\bigr)(a\_1-2).
$$



Graphs with prescribed numbers of k-cliques
What you are asking for is possible $f$-vectors of [Clique Complexes](https://en.m.wikipedia.org/wiki/Clique_complex). The [Kruskal-Katona Theorem](https://en.m.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem) characterizes $f$-vectors of simplicial complexes; so, it applies here but is no longer a complete characterization. Your problem is open according to [this paper](http://arxiv.org/pdf/1403.6210.pdf). The linked paper and references there in contain partial results.


**Edit:** This answer assumes the number of $k$-cliques of $G$ is $a\_k$ for $1 \leq k \leq n$ and $0$ for $k > n$. Another interpretation of the question would be that the number of $k$-cliques of $G$ is $a\_k$ for $1 \leq k \leq n$ and there is no restriction on the number of $k$-cliques for $k > n$. The OP has expressed interest in both variations in the comments.


