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Prellberg-Stanton Conjecture

The Prellberg-Stanton conjecture is the coefficient-positivity assertion

F_(n,m)(q)=(1-q)/((q^n;q)_m)[n; m]_q in N[[q]]

for integers 0<m<n, except when n is even and m=2 or m=n-2. Here the bracket is a q-binomial coefficient, (q^n;q)_m is a q-Pochhammer symbol, and membership in N[[q]] means that the formal power series has nonnegative integer coefficients. This is not an assertion that F_(n,m) is a polynomial.

Prellberg (2026) proved the conjecture by establishing positivity of the polynomial

R_(n,m)(q)=([m]_(q^(n+1)))/([n]_q)[n; m]_q

for the same pairs, where [j]_q is a q-bracket. The same method also proves the Reiner-Stanton-White conjecture for flag-orbit polynomials.

See also

q-Binomial Coefficient, q-Pochhammer Symbol

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References

Prellberg, T. "A Positivity Conjecture of Prellberg and Stanton." 8 Oct 2026. https://arxiv.org/abs/2610.26878.

Cite this as:

Weisstein, Eric W. "Prellberg-Stanton Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Prellberg-StantonConjecture.html

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