Counting Pattern-free Set Partitions II: Noncrossing and Other Hypergraphs

A (multi)hypergraph ${\cal H}$ with vertices in ${\bf N}$ contains a permutation $p=a_1a_2\ldots a_k$ of $1, 2, \ldots, k$ if one can reduce ${\cal H}$ by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to ${\cal H}_p=(\{i, k+a_i\}:\ i=1, \ldots, k)$. We formulate six conjectures stating that if ${\cal H}$ has $n$ vertices and does not contain $p$ then the size of ${\cal H}$ is $O(n)$ and the number of such ${\cal H}$s is $O(c^n)$. The latter part generalizes the Stanley–Wilf conjecture on permutations. Using generalized Davenport–Schinzel sequences, we prove the conjectures with weaker bounds $O(n\beta(n))$ and $O(\beta(n)^n)$, where $\beta(n)\rightarrow\infty$ very slowly. We prove the conjectures fully if $p$ first increases and then decreases or if $p^{-1}$ decreases and then increases. For the cases $p=12$ (noncrossing structures) and $p=21$ (nonnested structures) we give many precise enumerative and extremal results, both for graphs and hypergraphs.