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Abstract:
Linear recurrences (and, with them, ordinary generating functions)
are ubiquitous in combinatorics, as part of a broad general framework
that is well-studied and well-understood; in contrast, bilinear
recurrences such as
s_{n+4} = (s_{n+1} s_{n+3} + s_{n+2}^2) / s_{n}
are encountered far less often, and these encounters tend to be viewed
in isolation from one another.
In this talk I will describe some types of combinatorial objects
whose properties make them well-suited to a (nascent) general theory
of bilinear recurrence relations. In some interesting cases (e.g.,
the Somos-4 recurrence given above), algebra is one step ahead of
combinatorics, and we are temporarily in the unusual position of
being able to enumerate combinatorial objects for which we lack a
combinatorial description!
I will attempt to convince members of the audience that some basic
problems connected with bilinear recurrence relations are compelling
and accessible. If I succeed at this, I plan to organize a working
group that will jointly explore these problems over the next several
months.
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