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 S_n. We prove that if n is large enough, up to isomorphism and duality, the number of string C-groups of rank r for S_n (with r ≥ (n+3)/2) is the same as the number of string C-groups of rank r+1 for S_n+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 S_n with n odd, one can construct from them all string C-groups of rank (n + 3)/2 + k for S_n+k for any positive integer k. The classification of the string C-groups of rank r ≥ (n + 3)/2 for S_n is thus reduced to classifying string C-groups of rank r for S_2r−3. A consequence of this result is the complete classification of all string C-groups of S_n 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.