The gamma vector of a polynomial 
satisfying 
is the sequence
of coefficients 
in its unique expansion
 |
Here 
is the floor
function. The polynomial has symmetric coefficients,
and the displayed polynomials form a basis
for polynomials with this symmetry of degree at most
(Horiguchi et al. 2026).
For example, 
has gamma vector 
,
whereas 
has gamma vector 
.
Nonnegative polynomial coefficients therefore need
not imply nonnegative gamma coefficients.
For a 
-dimensional
simple polytope, write 
for the number of 
-dimensional faces, with 
. The polynomial 
is symmetric by the Dehn-Sommerville
equations. Horiguchi et al. (2026) prove that its gamma vector is nonnegative
for every partitioned permutohedron.
