泊车函数、斯米尔诺夫字与非交叉Chow多项式
Real-rootedness and interlacing for parking functions and Chow polynomials
- Stockholm University（斯德哥尔摩大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过转换泊车函数与斯米尔诺夫字的关系，结合递推关系证明了非交叉分拆格Chow多项式等的实根性，解决了相关猜想并得到多个多项式的实根性结论。
AI中文摘要：
我们通过将无约束泊车函数转换为有限字母表上的斯米尔诺夫字，并应用末字母交错递推关系，证明了非交叉分拆格的Chow多项式的实根性。我们还推导了关于峰和约束的三角递推关系，并确定无峰无约束下降多项式为Narayana多项式。对于Ehrenborg–Hetyei–Readdy的环面g-贡献，我们构造了一个固定行公共交错子，并证明了所有非负行和的实根性，由此特别可推出单个多项式的实根性；Q. Xiao最近通过不同的微分递推关系独立证明了这一结论，我们也给出了该单个结论的另一个有限Schur–Szegő卷积证明。这些结果证明了Xiao的猜想4.2以及Ehrenborg–Hetyei–Readdy的猜想11.2，对弱避免123模式的泊车函数具有相关推论。我们还证明了所有泊车函数上的像大小多项式，以及四类避免两个模式的泊车函数的上升和下降多项式的实根性。
英文摘要：
We prove real-rootedness for the Chow polynomials of the noncrossing partition lattices by transferring tieless parking functions to finite-alphabet Smirnov words and applying an interlacing-preserving transition of M.~Leander. We also derive a triangular recurrence for peaks and ties and identify the peakless-tieless descent polynomial as the Narayana polynomial. For the toric $g$-contributions of Ehrenborg--Hetyei--Readdy, we exhibit a fixed-row common interlacer and establish real-rootedness of all nonnegative row sums. Individual real-rootedness follows in particular; Q.~Xiao also proved it by a different differential recurrence. We also give a second proof of the individual statement, by finite Schur--Szegő convolution, that does not use the common interlacer. These results prove Conjecture~4.2 of Xiao and Conjecture~11.2 of Ehrenborg--Hetyei--Readdy, with consequences for weakly 123-avoiding parking functions. We also prove real-rootedness for the image-size polynomial on all parking functions and for the ascent and descent polynomials of four two-pattern-avoiding classes.
