论文标题
高级字符串C组的数量
The number of string C-groups of high rank
论文作者
论文摘要
如果$ g $是一个具有$ n $的及时级别$ n $的c-group级$ r \ geq（n+3）/2 $，则$ g $一定是对称组$ s_n $。我们证明，如果$ n $足够大，以等效和二元性，则$ s_n $的等级$ r $的字符串c组数量（带有$ r \ geq（n+3）/2 $）与$ s_ {n+1} $的等级$ r+1 $ r+1 $的字符串c组数量相同。该结果以及在其证明中使用的工具，尤其是等级和程度扩展，这意味着，如果一个人知道等级$（n+3）/2 $的字符串c组，$ n $ odd的$ s_n $ for $ n $ odd，则可以从它们中构建所有等级$（n+3）/2+k $的字符串c组，用于$ s_ {n+k} $ for n n+k $ for note protical inte $ k $。因此，$ s_n $的等级$ r \ geq（n+3）/2 $的字符串c组的分类减小为$ s_ {2r-3} $的字符串c groups c-groups c-groups。该结果的结果是将$ s_n $的所有字符串c组与等级$ n-κ$ for $κ\ in \ {1，\ ldots，6 \} $，当$ n \ geq2κ+3 $时，这扩展了先前已知的结果。等级$ n-κ$的字符串C组数量（$ n \ geq2κ+3 $）的此类分类为以下由$κ$索引的整数顺序，从$κ= 1 $：$ $：$ $：1,1,7,9,9,35,48）$ bude cequine cequine cequine cequine cenegerclopepedeclopepeclopepeclopepeclopedececlopedececlopedeclopepediaia是Intecyclopedia的新序列。它将作为序列编号A359367提供。
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 $n-κ$ for $κ\in\{1,\ldots,6\}$, when $n\geq 2κ+3$, which extends previously known results. The number of string C-groups of rank $n-κ$, with $n\geq 2κ+3$, of this classification gives the following sequence of integers indexed by $κ$ and starting at $κ= 1$: $$(1,1,7,9,35,48)$$ This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It will be available as sequence number A359367.
