Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A generalization of a theorem of Erné (2105.00711v1)

Published 3 May 2021 in math.CO

Abstract: Let $X$ be a finite set, $Z \subseteq X$ and $y \notin X$. Marcel Ern\'{e} showed in 1981, that the number of posets on $X$ containing $Z$ as an antichain equals the number of posets $R$ on $X \cup { y }$ in which the points of $Z \cup { y }$ are exactly the maximal points of $R$. We prove the following generalization: For every poset $Q$ with carrier $Z$, the number of posets on $X$ containing $Q$ as an induced sub-poset equals the number of posets $R$ on $X \cup { y }$ which contain $Qd + A_y$ as an induced sub-poset and in which the maximal points of $Qd + A_y$ are exactly the maximal points of $R$. Here, $Qd$ is the dual of $Q$, $A_y$ is the singleton-poset on $y$, and $Qd + A_y$ denotes the direct sum of $Qd$ and $A_y$.

Summary

We haven't generated a summary for this paper yet.