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Approximation and Combinatorics (1310)
Approximation et combinatoire
Dates: 20-24 April 2015 at CIRM (Marseille, France)      

Place : CIRM (Marseille Luminy, France)
Generating series for counting functions abound in combinatorics. Their properties (formal, analytic, algebraic, rational, D-finite, holonomic) provide information on the analytic and algebraic relations among the coefficients. These properties are often very hard to detect, and suitable techniques for this would be very welcome.

For instance, Bostan and Kauers have proven that the generating function of Gessel walks is algebraic, but the proof is very complicated and relies on a huge computer implementation. This is precisely a situation where the standard tool to prove algebraicity, the division theorem, is not applicable since the divisor series is not regular.

The workshop aims at understanding better the appearance of analytic or algebraic power series in combinatorics, to look out for new tools, and to compare them with D-finite and holonomic series.

On the other hand, certain combinatorial identities can be expressed through the equality of their generating functions, as is the case e.g. for the Rogers-Ramanujan identity. Bruschek, Mourtada and Schepers showed that one side of the RR identity appears as the Hilbert-Poincaré-series of an arc space, namely nilpotent arcs defined by x(t)²= 0. This suggests to interpret also the other side of the identity through arc spaces and then to prove the identity by methods of algebraic geometry. This, however, is still an open problem.