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%           FPSAC/SFCA '94, May 23-27, 1994   at DIMACS
%                     Problem Session
%   LaTeX file of problem  #2  submitted by David M. Jackson (May 24, 1994)
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{FPSAC/SFCA '94}\\
{PROBLEM SESSION}\\
{Submission received May 24, 1994}\\[.2in]
{\bf AN IDENTITY FOR MAPS ON ORIENTABLE SURFACES}\\[.1in]
David M.\ Jackson\\[.2in]
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Let $M(u,x,y,z)$ be the generating series for rooted maps on
orientable surfaces, where $u$ marks genus, $x$ marks number of
vertices, $y$ marks number of faces, and $z$ marks number of edges.
Let $Q(u,x,y,z)$ be the generating function for rooted
quadrangulations (face 4-regular maps) on orientable surfaces.
Prove combinatorially that
  $$ Q(u^2,x,y,z) = \frac{1}{2}\left( M(4u^2,x+u,x,yz^2) + M(4u^2,
     x-u,x,yz^2)\right). $$
\noindent See {\em Trans.\ Amer.\ Math.\ Soc.}\ {\bf 322} (1990), 343--363.

The result is stated on p.\ 358 as Corollary 5.2, and a small example is given
 on
p. 361. In my talk I referred to $M$ and $Q$ as {\em genus series}.

The present proof is a hard character theoretic one. It may be that this
result can be proved by surgical operations of pasting, cutting and glueing
on the surface, and that the algorithm will be of wider application.
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