Creation operators act on symmetric functions to build Schur functions, Hall–Littlewood polynomials, and related symmetric functions one row at a time. Haglund, Morse, Zabrocki, and others have studied more general symmetric functions $H_{\alpha }$, $C_{\alpha }$, and $B_{\alpha }$ obtained by applying any sequence of creation operators to 1. We develop new combinatorial models for the Schur expansions of these and related symmetric functions using objects called abacus-histories. These formulas arise by chaining together smaller abacus-histories that encode the effect of an individual creation operator on a given Schur function. We give a similar treatment for operators such as multiplication by $h_m$, $h_m^{\perp }$, ω, etc., which serve as building blocks to construct the creation operators. We use involutions on abacus-histories to give bijective proofs of properties of the Bernstein creation operator and Hall–Littlewood polynomials indexed by three-row partitions.

创建运算符对对称函数起作用，以一次建立一行Schur函数，Hall–Littlewood多项式以及相关的对称函数。Haglund，Morse，Zabrocki等人研究了更一般的对称函数$H_{\alpha }$， $C_{\alpha }$和 ${乙}_{\alpha }$通过将任意创建运算符序列应用于1可以得到结果。我们使用称为算盘历史的对象为这些和相关对称函数的Schur展开开发了新的组合模型。这些公式是通过将较小的算盘历史记录链接在一起而产生的，这些历史记录编码单个创建算符对给定Schur函数的影响。我们对运算符给予类似的处理，例如乘以$H_{米}$， $H_{米}^{\perp }$，ω等，它们用作构造创建运算符的构建块。我们使用算盘历史上的对合来对伯恩斯坦创建算子和由三行分区索引的Hall–Littlewood多项式的性质给出双射证明。