Abstract
The q-multinomial coefficient, a classical object in enumerative combinatorics, counts permutations of multisets weighted by the number of inversions, with a single deformation parameter q. We introduce the twisted multinomial coefficient, in which each inversion between letters i and j carries a pair-dependent weight ω_ij determined by a skew-symmetric matrix Ω. In general, no closed-form evaluation is known. Our main result is that under a natural structural condition on Ω - predecessor-uniformity (ω_ij = q_j for all i<j) - the twisted multinomial factorizes as a product of Gaussian (q-deformed) binomials with site-dependent parameters: kk₁,…,k_m_Ω = Π_jℓ_jk_j_q_j where ℓ_j = k₁+…+k_j. This extends the standard product formula for the q-multinomial from a single parameter q to m-1 independent parameters. The identity is purely combinatorial: it holds for arbitrary q_j ∈ C{0} without any algebraic constraints. We were led to this identity by studying pilot state preparation in Hamiltonian Decoded Quantum Interferometry (HDQI), a recently proposed quantum algorithm for preparing Gibbs and ground states. As an application, we show that the factorization yields an exact matrix product state (MPS) of bond dimension k+1 for the expansion coefficients of h^k in a twisted algebra. We further show that the same site matrices deliver an exact MPS of bond dimension deg(P)+1 for the expansion coefficients of P(h), for any polynomial P, via a polynomial-dependent right boundary vector.