A polynomial P ∈ C[z1, . . . , zd] is strongly Dd-stable if P has no zeroes in the closed unit polydisc D d . For such a polynomial define its spectral density function as SP (z) = P(z)P(1/z) −1 . An abelian square is a finite string of the form ww0 where w0 is a rearrangement of w. We examine a polynomial-valued operator whose spectral density function’s Fourier coefficients are all generating functions for combinatorial classes of con- strained finite strings over an alphabet of d characters. These classes generalize the notion of an abelian square, and their associated generating functions are the Fourier coefficients of one, and essentially only one, L2 (T d)-valued oper- ator. Integral representations and asymptotic behavior of the coefficients of these generating functions and a combinatorial meaning to Parseval’s equation are given as consequences.