q-Zeta Polynomial

The -zeta polynomial of a nonempty finite partially ordered set with a height function is the polynomial in over characterized by

where , is the q-bracket, and the sum is over weakly increasing sequences in . The height function takes nonnegative integer values and satisfies whenever in (Chapoton 2026, pp. 3-5).

The polynomial degree of over is . Its specialization at is the classical zeta polynomial , so it is a q-analog of that invariant. The polynomial degree can drop under this specialization. Although each value at for has nonnegative integer coefficients as a polynomial in , this does not imply nonnegative coefficients in every expansion of (Chapoton 2026, pp. 5-6).

For a three-element chain whose successive heights are 0, 1, and 2, the defining sum gives

At this becomes , the zeta polynomial of the three-element chain.