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\specs{P37 (comment)}{18(2)}{2012}

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\title{\bf A Comment on paper P37 of volume 18(2)}

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\author{Qing-Hu Hou \\
\small Center for Combinatorics\\[-0.8ex]
\small Nankai University\\[-0.8ex]
\small Tianjin P.R. China \\
\small\tt hou@nankai.edu.cn\\
}

\date{\dateline{Oct 30, 2012}{Jan 4, 2013}{Jan 7, 2013}}


\begin{document}
\maketitle

Conjecture 10 follows from the following recurrence relation. Suppose that $a_h=2a'_h$ is even. Then
\[
  H_{a_1,\dots,a_s}
  = \begin{cases}
  H_{a_1,\dots,a_{h-1}, a'_{h}, a_{h+1},\dots,a_s} - H_{a_1,\dots,a_{h-1}, a_{h}+a_{h+1}, a_{h+2}, \dots, a_s} \\
  \qquad + H_{a_1,\dots,a_{h-2},a_{h-1}+a'_h, a_{h+1}, \dots, a_s}, & 1 < h < s, \\[15pt]
   H_{a'_1, a_2,\dots,a_s} - H_{a_1+a_2, a_3, \dots, a_s} & 1=h<s, \\[10pt]
   H_{a_1,\dots,a_{s-1}, a'_s} - z^{a'_s} H_{a_1,\dots,a_{s-1}} + H_{a_1,\dots,a_{s-2},a_{s-1}+a'_s},
    & 1 < h = s, \\[10pt]
   H_{a'_1} - z^{a'_1}, & h=s=1.
   \end{cases}
\]
\end{document}