= Solution
For a null momentum, use <light cone gauge> to remove potential components along one null direction; the free field equations remove the remaining longitudinal components. The physical polarizations of a <massless p-form gauge field> are therefore antisymmetric tensors on the $D-2$ transverse directions, transforming under the rotational <little group> $SO(D-2)$. Choosing $p$ distinct transverse indices gives
$$
\boxed{n_p(D)=\binom{D-2}{p}.}
$$
This already takes account of the reducibility of the higher-form gauge parameter; naïvely subtracting just one gauge-parameter component count would not give the right answer. No self-duality constraint is being imposed.
For <dimensional reduction> to four dimensions, retain the zero modes on a flat $(D-4)$-torus, without fluxes or projections. Write $d=D-4$ and split tensor indices into spacetime indices $\mu$ and internal indices $i$. A component with $r$ spacetime indices is a four-dimensional $r$-form, with $p-r$ internal indices and hence multiplicity $\binom d{p-r}$. The complete field decomposition is
$$
A_p\longrightarrow\bigoplus_{r=0}^{\min(p,4)}\binom d{p-r}\ A_r^{(4)}.
$$
Use the convention that a binomial coefficient vanishes when its lower entry lies outside its range. In four dimensions the scalar, vector and two-form have respectively $1,2,1$ local polarizations, while the three- and four-form potentials have none. Hence <toroidal reduction of a p-form gauge field> gives
$$
\begin{aligned}
n_{\rm reduced}&=\binom dp+2\binom d{p-1}+\binom d{p-2}\\
&=\sum_r\binom2r\binom d{p-r}=\binom{d+2}p=\binom{D-2}p.
\end{aligned}
$$
The middle identity follows by comparing the coefficient of $t^p$ in $(1+t)^2(1+t)^d=(1+t)^{d+2}$. Therefore
$$
\boxed{n_{\rm reduced}=n_p(D).}
$$
A four-dimensional two-form may instead be counted as one scalar by <two-form scalar duality>, with the same result. Higher-form zero modes can describe nondynamical flux data, which are distinct from these propagating degrees of freedom. Nonzero <Kaluza-Klein modes> form massive lower-dimensional multiplets with longitudinal components supplied by the associated gauge fields; the displayed matching concerns the massless zero-mode sector.
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