Enumerative and combinatorial properties of Dyck partitions

Francesco Brenti

MIT and Universita' di Roma "Tor Vergata"

March 14,
refreshments at 3:45pm


The purpose of this talk is to introduce a new class of (skew) integer partitions and study their enumerative and combinatorial properties. This class is closely related to Dyck paths, and plays a fundamental role in the computation of certain Kazhdan-Lusztig polynomials of the symmetric group related to Young's lattice.

More precisely, after giving the definitions and some examples, I will state the fundamental combinatorial properties of Dyck partitions, including a combinatorial characterization. I will then illustrate a bijection that allows the enumeration of Dyck partitions. This bijection is based on a statistic on integer partitions that seems to be new. Finally, I will briefly explain the connection between Dyck partitions and the Kazhdan-Lusztig polynomials, and, using this connection, I will derive some new identities for these polynomials. I will conclude by discussing some possible lines of further investigation, and in particular possible analogues of Dyck partitions for other Coxeter groups.

Speaker's Contact Info: brenti(at-sign)math.mit.edu

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