The number of string C-groups of high rank

If Gis a transitive group of degree nhaving a string C-group of rank r=( n + 3)/2, then Gis necessarily the symmetric group Sn. We prove that if nis large enough, up to isomorphism and duality, the number of string C-groups of rank rfor Sn(with r=( n + 3)/2) is the same as the number of string C-groups of rank r+ 1for 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)/2for Snwith nodd, one can construct from them all string C-groups of rank ( n + 3)/2 + kfor Sn+kfor any positive integer k. The classification of the string C-groups of rank r=( n + 3)/2for Snis thus reduced to classifying string C-groups of rank rfor S2r-3. A consequence of this result is the complete classification of all string C-groups of Snwith rank n -.for..{1,..., 7}, when n = 2.+3, which extends previously known results. The number of string C-groups of rank n -., with n = 2. + 3, of this classification gives the following sequence of integers indexed by.and starting at. = 1: