The asymptotic number of planar, slim, semimodular lattice diagrams (Q304178)

The author gives interesting results on the number \(N_{\mathrm{ssd}}(n)\) of slim, semimodular diagrams of size \(n\). More precisely, it is proved that there exists a positive constant \(C < 1\) such that \(N_{\mathrm{ssd}}(n)\) is asymptotically \(C \cdot 2^n\), that is, \(\lim_{n\rightarrow \infty}(N_{\mathrm{ssd}}(n)/2^n)=C\). This result allows us to know many ways the members of two composition series in a group can intersect each other, provided that there are exactly \(n\) intersections and that we make a distinction between the first composition series and the second.