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).

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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