AI 中文总结
解决阿姆德贝汉等人关于记录组合是否有类似理论的开放问题，证明具有特定记录组合的\(\{1,\dots,2n\}\)交错排列数公式，该数是芽对称函数提升到非交换对称函数展开系数，且对特定芽序列有类似细化，结果由AxiomProver在Lean中验证。
AI 中文摘要
阿姆德贝汉、沙雷希安和斯坦利最近证明，在分区艾森斯坦级数理论中出现的函数\(\varphi\)可对具有给定“记录”分区的\(\{1,\dots,2n\}\)的交错排列进行计数，他们还询问是否存在关于记录组合的类似理论，暗示了非交换对称函数的作用。本文通过证明具有记录组合\((\alpha_1,\dots,\alpha_\ell)\)的\(\{1,\dots,2n\}\)的交错排列数为\(\prod_{j=1}^{\ell}\binom{2s_j - 1}{2\alpha_j - 1}E_{2\alpha_j - 1}\)来解决他们的开放问题，其中\(s_j=\alpha_1+\dots+\alpha_j\)，\(E_k\)是欧拉数。记录组合列出了在每个从左到右的最大值（除第一个外）之前切割\(a_1a_3\dots a_{2n - 1}\)得到的因子长度。这些数是度为\(n\)、种子为\(\sec(\sqrt{t}\,)\)的芽对称函数自然提升到非交换对称函数并展开为第一类非交换幂和乘积的系数。对于由指数公式给出种子的每个芽序列都有类似的细化。AxiomProver在Lean中自主生成并验证了本文的结果。
英文摘要
Amdeberhan, Shareshian, and Stanley recently proved that a function $φ$ 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 $(α_1,\dots,α_\ell)$ is \[ \prod_{j=1}^{\ell}\binom{2s_j-1}{2α_j-1}E_{2α_j-1}, \] where $s_j=α_1+\dots+α_j$, $E_k$ 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.
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