KK-like relations of $\alpha^{\prime}$ corrections to disk amplitudes

Inspired by the definition of color-dressed amplitudes in string theory, we define analogous {\it color-dressed permutations} replacing the color-ordered string amplitudes by their corresponding permutations. Decomposing the color traces into symmetrized traces and structure constants, the color-dressed permutations define {\it BRST-invariant permutations}, which we show are elements of the inverse Solomon descent algebra and we find a closed formula for them. We then present evidence that these permutations encode KK-like relations among the different $\alpha'$ corrections to disk amplitudes refined by their motivic MZV content. In particular, the number of linearly independent amplitudes at a given ${\alpha^{\prime}}$ order and motivic MZV content is given by (sums of) Stirling cycle numbers. In addition, we show how the superfield expansion of BRST invariants of the pure spinor formalism corresponding to ${\alpha^{\prime}}^2 f_2$ corrections is encoded in the descent algebra.