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Topic: Difficult expression, possible to simplify to (n+1)^(n-1)?
Replies: 1   Last Post: Feb 28, 1998 3:22 AM

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Dan Aaronson

Posts: 10
Registered: 12/7/04
Difficult expression, possible to simplify to (n+1)^(n-1)?
Posted: Feb 26, 1998 4:37 AM
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During work om my math thesis i have come across a strange expression
that it seems to
be possible to simplify quite a lot.
When you count the number of complete exceptional sequences over the
hereditary algebra An you get the expression I for convenience have
attached in a Sci.Word file.
The original hypothesis was however that I would get (n+1)^(n-1), and
when you test the
two expressions in maple you find that the difference is 0 at least up
to n=11.
I would like to get a proof that the two expressions are equal for all
positive integers,
and, no, you can not use the binomial rule because the terms x and y in
(x+y)^n change
for each term inside the expression.

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\documentclass{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{graphicx}
\usepackage{amsmath}

%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Created=Thu Feb 26 10:27:30 1998}
%TCIDATA{LastRevised=Thu Feb 26 10:31:26 1998}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="General\Blank Document">}
%TCIDATA{CSTFile=LaTeX article (bright).cst}

\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}

\begin{document}


$\Sigma _{i=1}^{n}\binom{n}{i}(n-i+1)^{n-i-1}(i+1)^{i}$

\end{document}

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