I will discuss three combinatorial problems with the same solution.

In 2002, R. Suter defined a striking cyclic symmetry of order n+1 on the subposet of Young’s lattice consisting of the 2^n partitions with largest hook at most n. These partitions naturally arise from D. Peterson's parametrization of abelian ideals of a Borel subalgebra using the affine Weyl group (as told by B. Kostant); the cyclic symmetry comes from the fact that the affine Dynkin diagram in type A is a cycle.

Problem 1. Describe the orbit structure of Suter's cyclic symmetry.
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Problem 2. Show that the sweep map is a bijection on Dyck(a,b).

Problem 3. With these assumptions, find an assignment of starting hours for the tasks so that the workload throughout the day is constant.

This is based on joint work with Hugh Thomas.

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