Somos sequences and bilinear combinatorics

James Propp

University of Wisconsin and MIT

October 18,
refreshments at 3:45pm


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

sn+4 = (sn+1 sn+3 + sn+22) / sn
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.

(Most of the lecture should be accessible to advanced undergraduates.)

Speaker's Contact Info: propp(at-sign)

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