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The number $(n+1)^(n-1)$ is familiar to combinatorialists as the
number of rooted forests on $n$ vertices. The Catalan number $C_n$
is another familiar friend, and both of these have well-known
"$q$-analogs" that enumerate combinatorial objects according to
various "weights." In recent years these same numbers made a
surprising appearance in a geometric context. The theorems that
establish the connection have only recently been proven. In the
geometric context we get "$q,t$-analogs" with a second parameter
$t$. On the combinatorial side this means there should be a second
set of weights to go with the ones that were known before. This
raises a number of interesting problems that are mostly still
unsolved.
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