The number of string C-groups of high rank

TL;DR

This paper investigates the structure and classification of string C-groups of high rank for symmetric groups, establishing a recursive approach to classify them and providing a new integer sequence related to their counts.

Contribution

It introduces a method to classify high-rank string C-groups of symmetric groups by reducing the problem to smaller cases, and fully classifies those with rank close to the degree.

Findings

Number of string C-groups of rank r for S_n matches that of rank r+1 for S_{n+1} for large n.

Complete classification of string C-groups of rank n-κ for κ=1 to 6 when n is large enough.

New integer sequence A359367 enumerates these string C-groups, extending known results.

Abstract

If $G$ is a transitive group of degree $n$ having a string C-group of rank $r\geq (n+3)/2$, 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 $r\geq (n+3)/2$) 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 $r\geq (n+3)/2$ 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 $n-\kappa$ for $\kappa\in\{1,\ldots,6\}$, when $n\geq 2\kappa+3$, which extends previously known results. The number of string C-groups of rank $n-\kappa$, with $n\geq 2\kappa +3$, of this classification gives the following sequence of integers indexed by $\kappa$ and starting at $\kappa = 1$: $$(1,1,7,9,35,48)$$ This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It will be available as sequence number A359367.