Summary: Lifted generalized permutahedra and
Generalized permutahedra are the polytopes obtained from the permutahe-
dron by changing the edge lengths while preserving the edge directions, pos-
sibly identifying vertices along the way. We introduce a lifting construction
for these polytopes, which turns an n-dimensional generalized permutahedron
into an (n + 1)-dimensional one. We prove that this construction gives rise
to Stasheff's multiplihedron from homotopy theory, and to the more general
nestomultiplihedra, answering two questions of Devadoss and Forcey.
We construct a subdivision of any lifted generalized permutahedron whose
pieces are indexed by compositions. The volume of each piece is given by a
polynomial whose combinatorial properties we investigate. We show how this
composition polynomial arises naturally in the polynomial interpolation of an
exponential function. We prove that its coefficients are positive integers, and
present evidence suggesting that they may also be unimodal.
San Francisco State University, San Francisco, CA, USA, email@example.com.