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{\bf Carl W. Lee}
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{\bf Sweeping the ${\bf cd}$-Index and the Toric $h$-Vector}
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We derive formulas for the ${\bf cd}$-index and the toric $h$-vector of a
convex polytope $P$ from a sweeping by a hyperplane. These arise from
interpreting the corresponding $S$-shelling of the dual of $P$. We
describe a partition of the faces of the complete truncation of $P$ to
reflect explicitly the nonnegativity of its ${\bf cd}$-index and what its
components are counting. One corollary is a quick way to compute the
toric $h$-vector directly from the ${\bf cd}$-index that turns out to be
an immediate consequence of formulas of Bayer and Ehrenborg. We also
propose an ``extended toric'' $h$-vector that fully captures the
information in the flag $h$-vector.
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