  A notion of cyclic descents on standard Young tableaux (SYT) of rectangular
shape was introduced by Rhoades, and extended to certain skew shapes by the
last two authors. The cyclic descent set restricts to the usual descent set
when the largest value is ignored, and has the property that the number of SYT
of a given shape with a given cyclic descent set $D$ is invariant under cyclic
shifts of the entries of $D$. Following these results, the existence of cyclic
descent sets for standard Young tableaux of any skew shape other than a ribbon
was conjectured by the authors, and recently proved by Adin, Reiner and
Roichman. Unfortunately, the proof does not provide a natural definition of the
cyclic descent set for a specific tableau.
  In this paper we explicitly describe cyclic descent sets and resulting
generating functions for SYT of (possibly skew) shapes which either have
exactly two rows or are near-hooks, i.e., are one cell away from a hook. Our
definition provides a constructive combinatorial proof of the existence of
cyclic descent sets for these shapes, and coincides with that of Rhoades for
two-row rectangular shapes. We also show that cyclic descent sets for near-hook
shaped tableaux are unique.
