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F50-05: Determinantal and Recursive Methods in Enumeration

Principal Investigator: Christian Krattenthaler

This project part aims at developing methods for solving (exact and asymptotic) enumerative problems connected with rhombus tilings, tableaux, plane partitions and related objects and algorithms, which so far have remained elusive or have not been attacked at all. On the exact side, several open conjectures and problems on the enumeration of rhombus tilings of hexagons with holes will be in the focus. Partly building on results obtained on this project goal, (asymptotic) correlations of holes in regions with free boundary will be computed, going beyond the (so far) isolated result by Ciucu and Krattenthaler. Parallels with laws of electrostatics for charges under the presence of a conductor are expected. A further goal is the exact and asymptotic analysis of the complexity (measured in number of steps) of Robinson--Schensted--Knuth-type algorithms for tableaux and plane partitions. Motivated by the steadily increasing importance of recurrences in proofs of enumeration formulas, the fundamental problem of finding effective results for the verification of a recurrence by a verification of sufficiently many instances will be attacked.

Christian Krattenthaler