Record compositions of alternating permutations and noncommutative symmetric functions

Amdeberhan, Shareshian, and Stanley recently proved that a function $\varphi$ arising in the theory of partition Eisenstein series counts the alternating permutations of $\{1,\dots,2n\}$ with a given `record' partition, and they asked whether there is a similar theory for record compositions, suggesting a role for noncommutative symmetric functions. Here we solve their open problem by showing that the number of alternating permutations of $\{1,\dots,2n\}$ with record composition $(\alpha_1,\dots,\alpha_\ell)$ is
\[ \prod_{j=1}^{\ell}\binom{2s_j-1}{2\alpha_j-1}E_{2\alpha_j-1}, \]
where $s_j=\alpha_1+\dots+\alpha_j$, $E_k$ is an Euler number, and the record composition of $w=a_1a_2\dots a_{2n}$ (so $a_1>a_2<a_3>\dotsb$) lists the factor lengths obtained by cutting $a_1a_3\dots a_{2n-1}$ before each left-to-right maximum other than the first. These numbers are the coefficients of a natural lift of the degree-$n$ sprout symmetric function with seed $\sec(\sqrt{t}\,)$ to noncommutative symmetric functions, expanded in products of noncommutative power sums of the first kind. An analogous refinement holds for every sprout sequence whose seed is given by the exponential formula. AxiomProver autonomously produced and verified the results in this paper in Lean.