R\'enyi公式的推广

An Extension of the R\'enyi Formula

中文关键词:  标号超图  $(p,~q)$超单圈  匀称$(k+1)$秩  R\'enyi公式

英文关键词:Labeled hypergraph  $(p,~q)$-unicycles  $(k+1)$-uniform  R\'enyi formula

中文摘要:

      在本文,作为著名的R\'enyi公式(其刻画了标号连通单圈图的计数显式)的自然推广,我们研究了标号匀称$(k+1)$秩$(p,~q)$超单圈的计数问题,给出了如下的计数显式:$$U_{p,~q}^{(k+1)}=\begin{cases} \frac{p!}{2[(k-1)!]^q}\cdot\sum_{t=2}^q \frac{q^{q-t-1}\cdot sgn(tk-2)}{(q-t)!}, & p=qk, \\ 0,& p\neq qk, \end{cases}$$其中$k,~p,~q$均为正整数.

英文摘要:

      In this paper, as a natural extension of the R\'enyi formula which counts labeled connected unicyclic graphs, we present a formula for the number of labeled $(k+1)$-uniform $(p,~q)$-unicycles as follows: $$U_{p,~q}^{(k+1)}=\begin{cases} \frac{p!}{2[(k-1)!]^q}\cdot \sum_{t=2}^q \frac{q^{q-t-1}\cdot {\rm sgn}(tk-2)}{(q-t)!}, & p=qk, \\ 0, & p\neq qk, \end{cases}$$ where $k,~p,~q$ are positive integers.