  Partite, $3$-uniform hypergraphs are $3$-uniform hypergraphs in which each
hyperedge contains exactly one point from each of the $3$ disjoint vertex
classes. We consider the degree sequence problem of partite, $3$-uniform
hypergraphs, that is, to decide if such a hypergraph with prescribed degree
sequences exists. We prove that this decision problem is NP-complete in
general, and give a polynomial running time algorithm for third almost-regular
degree sequences, that is, when each degree in one of the vertex classes is $k$
or $k-1$ for some fixed $k$, and there is no restriction for the other two
vertex classes. We also consider the sampling problem, that is, to uniformly
sample partite, $3$-uniform hypergraphs with prescribed degree sequences. We
propose a Parallel Tempering method, where the hypothetical energy of the
hypergraphs measures the deviation from the prescribed degree sequence. The
method has been implemented and tested on synthetic and real data. It can also
be applied for $\chi^2$ testing of contingency tables. We have shown that this
hypergraph-based $\chi^2$ test is more sensitive than the standard $\chi^2$
test. The extra sensitivity is especially advantageous on small data sets,
where the proposed Parallel Tempering method shows promising performance.

Combinatorics;Discrete Mathematics