The Reiner-Stanton-White conjecture asserts nonnegativity of the coefficients of orbit-counting polynomials for flags
of finite field subspaces (Reiner et al. 2004,
p. 46). For a composition 
of 
and a positive divisor 
of 
, the free-orbit polynomial is
![O_(n,k)(q)=1/([n]_q)sum_(d|GCD(n,k_1,...,k_r))mu(d)[N/d; k_1/d,...,k_r/d]_(q^d),](/images/equations/Reiner-Stanton-WhiteConjecture/NumberedEquation1.svg) |
where the bracket is a q-multinomial coefficient, ![[n]_q](/images/equations/Reiner-Stanton-WhiteConjecture/Inline5.svg)
is a q-bracket, 
is the Moebius function, and 
is the greatest common
divisor. For prime power 
, it counts free orbits of the multiplicative quotient 
on flags in 
with successive dimension increments 
, ..., 
. Those orbits have size ![[n]_q](/images/equations/Reiner-Stanton-WhiteConjecture/Inline13.svg)
.
Prellberg (2026) proved that 
is a polynomial with
nonnegative integer coefficients. The other
prescribed orbit sizes follow by substitution and summation, resolving the conjecture.
This is a positivity theorem for flag-orbit polynomials, not for arbitrary cyclic
actions. Prellberg (2026) credits ChatGPT with assistance in exploring mathematical
arguments.
