Analyzing this one has interesting applications to the theory of random stable matching markets, as described in our paper.

This sequence is the best known lower bound on the maximum number of stable matchings on n men and n women. An analysis of the growth of this sequence was known, but never published. With some help from Peter Shor this analysis is finally available here.

One way to prove upper bounds on the worst-case maximum number of stable marriages is to prove upper bounds on the number of antichains in "tangled \(n \times n\) grid" posets, as observed by Dömötör Pálvölgyi. There is a simple tangled grid poset whose number of antichains is exactly A116078, as we show here. This sequence grows like \( \sqrt{n} 4^n \). This by itself disproves Pálvölgyi's conjecture that tangled \(n \times n\) grids have at most \(4^n\) antichains, and a modification of the analysis I gave for A005154 here actually proves that tangled grids can have up to \( 4.17^n \) antichains.

A lot of interesting combinatorial questions arise from investigating sets of linear orders over which majority voting never produces cycles.