Gorsky and Negut introduced operators ${\mathrm{Q}}_{m,n}$ on symmetric functions and conjectured that, in the case where m and n are relatively prime, the expression ${\mathrm{Q}}_{m,n}(1)$ is given by the Hikita polynomial ${\mathrm{H}}_{m,n}[X;q,t]$. Later, Bergeron-Garsia-Leven-Xin extended and refined the conjectures of ${\mathrm{Q}}_{m,n}(1)$ for arbitrary m and n which we call the Extended Rational Shuffle Conjecture. In the special case ${\mathrm{Q}}_{n+1,n}(1)$, the Rational Shuffle Conjecture becomes the Shuffle Conjecture of Haglund-Haiman-Loehr-Remmel-Ulyanov, which was proved in 2015 by Carlsson and Mellit as the Shuffle Theorem. The Extended Rational Shuffle Conjecture was later proved by Mellit as the Extended Rational Shuffle Theorem. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of ${\mathrm{Q}}_{m,n}(1)$ in certain special cases. Leven gave a combinatorial proof of the Schur function expansion of ${\mathrm{Q}}_{2,2n+1}(1)$ and ${\mathrm{Q}}_{2n+1,2}(1)$. In this paper, we explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of ${\mathrm{Q}}_{m,n}(1)$. Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of ${\mathrm{Q}}_{m,n}(1)$ in the case where m or n equals 3.

Gorsky和Negut引入了运营商 ${\mathrm{问}}_{米，ñ}$关于对称函数，并推测在m和n相对质数的情况下，表达式${\mathrm{问}}_{米，ñ}（\mathrm{1个}）$ 由Hikita多项式给出 ${\mathrm{H}}_{米，ñ}[X;q，Ť]$。后来，Bergeron-Garsia-Leven-Xin扩展并完善了${\mathrm{问}}_{米，ñ}（\mathrm{1个}）$对于任意的m和n，我们称其为扩展有理随机混搭猜想。在特殊情况下${\mathrm{问}}_{ñ+\mathrm{1个}，ñ}（\mathrm{1个}）$，Rational Shuffle Conjecture成为Haglund-Haiman-Loehr-Remmel-Ulyanov的Shuffle猜想，这在2015年由Carlsson和Mellit证明为Shuffle定理。扩展有理随机混搭猜想后来被Mellit证明为扩展有理随机混搭定理。本文的主要目的是研究在Schur函数展开时出现的系数的组合。${\mathrm{问}}_{米，ñ}（\mathrm{1个}）$在某些特殊情况下。Leven给出了Schur函数展开的组合证明${\mathrm{问}}_{2，2ñ+\mathrm{1个}}（\mathrm{1个}）$ 和 ${\mathrm{问}}_{2ñ+\mathrm{1个}，2}（\mathrm{1个}）$。在本文中，我们探索了在Schur函数展开时出现的系数的组合对称性。${\mathrm{问}}_{米，ñ}（\mathrm{1个}）$。特别是，我们研究了钩形Schur函数系数，以及${\mathrm{问}}_{米，ñ}（\mathrm{1个}）$在m或n等于3。