There is a natural notion of refinement for many CSP's. We formulate and prove a refinement of the aforementioned CSP arising from tracking the cyclic descent type of a word in addition to its content. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective.
A building block of our argument involves cyclic sieving for shifted
subset sums, which also appeared in Reiner-Stanton-White. We give an
alternate, largely bijective proof of a refinement of this result by
extending some ideas of Wagon-Wilf (1994).
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