How many ways can two composition series intersect?

Let H → and K → be finite composition series of length h in a group G . The intersections of their members form a lattice CSL ( H → , K → ) under set inclusion. Our main result determines the number N ( h ) of (isomorphism classes) of these lattices recursively. We also show that this number is asymptotically h ! / 2 . If the members of H → and K → are considered constants, then there are exactly h ! such lattices. on recent results of Czédli and Schmidt, first we reduce the problem to lattice theory, concluding that the duals of the lattices CSL ( H → , K → ) are exactly the so-called slim semimodular lattices, which can be described by permutations. Hence the results on h ! and h ! / 2 follow by simple combinatorial considerations. The combinatorial argument proving the main result is based on Czédli’s earlier description of indecomposable slim semimodular lattices by matrices.