The cyclic sieving phenomenon is a relation among a finite set, a cyclic group action, and evaluations of
a generating polynomial. More precisely, let a cyclic group 
of group order 
act on a finite set 
, and let 
be a generating polynomial
for a statistic on 
. The triple 
exhibits the cyclic sieving phenomenon if, for every
divisor 
of 
and every primitive 
th root of unity 
,
 |
Thus evaluations of 
at roots of unity count
the fixed points of corresponding elements of the
group action (Reiner et al. 2004).
Armstrong (2026) studied multisets of fixed cardinal number whose elements have multiplicity less
than a bound 
.
When a cyclic group acts by rotating the underlying
labels, the associated bounded q-binomial
coefficient gives cyclic sieving whenever the relevant rotation order is relatively
prime to 
.
This simultaneously generalizes the cases of subsets and
unrestricted multisets. Specializing at certain roots
of unity also connects these polynomials with
the two-denomination coin problem.
A related positivity question for flags of finite field subspaces is the Reiner-Stanton-White conjecture. Prellberg (2026) proved nonnegativity of its orbit-polynomial coefficients. This concerns coefficients of orbit-counting polynomials, rather than a new assertion about evaluations at roots of unity.
