The number of string C-groups of high rank

If G is a transitive group of degree n having a string C-group of rank r≥(n+3)/2, then G is necessarily the symmetric group Sn. We prove that if n is large enough, up to isomorphism and duality, the number of string C-groups of rank r for Sn (with r≥(n+3)/2) is the same as the number of string C-groups of rank r+1 for Sn+1. This result and the tools used in its proof, in particular the rank and degree extension, imply that if one knows the string C-groups of rank (n+3)/2 for Sn with n odd, one can construct from them all string C-groups of rank (n+3)/2+k for Sn+k for any positive integer k. The classification of the string C-groups of rank r≥(n+3)/2 for Sn is thus reduced to classifying string C-groups of rank r for S2r−3. A consequence of this result is the complete classification of all string C-groups of Sn with rank n−κ for κ∈{1,…,7}, when n≥2κ+3, which extends previously known results. The number of string C-groups of rank n−κ, with n≥2κ+3, of this classification gives the following sequence of integers indexed by κ and starting at κ=1:(1,1,7,9,35,48,135) This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It is available as sequence number A359367.