Unification, refinements and companions of generalisations of Schur's theorem

A generalisation of a partition theorem of Andrews to overpartitions

In 1969, Andrews proved a theorem on partitions with difference conditions which generalises Schur's celebrated partition identity. In this paper, we generalise Andrews' theorem to overpartitions. The proof uses q-differential equations and recurrences.

A generalisation of a second partition theorem of Andrews to overpartitions

In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur's celebrated partition identity (1926). Andrews' two generalisations of Schur's theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In a recent paper, the author generalised the first of these theorems to overpartitions, using a new technique which consists in going back and forth between $q$-difference equations on generating functions and recurrence equations on thei…

The method of weighted words revisited

Alladi and Gordon introduced the method of weighted words in 1993 to prove a refinement and generalisation of Schur's partition identity. Together with Andrews, they later used it to refine Capparelli's and Göllnitz' identities too. In this paper, we present a new variant of this method, which can be used to study more complicated partition identities, and apply it to prove refinements and generalisations of three partition identities. The first one, Siladić's theorem (2002), comes from vertex operator algebras. The second one, a conjectural identity of Primc (1999), comes from crystal base th…

A–D型经典群特征标的第九变体与Littlewood恒等式

Ninth variation of classical group characters of type A-D and Littlewood identities

本文推广经典李群特征标与有理Schur函数，用任意多项式序列替代阶乘幂，推导Littlewood型恒等式并获得新的Jacobi–Trudi与Nägelsbach–Kostka恒等式。

关于整数中的乘积Schur三元组

On product Schur triples in the integers

本文研究了整数中乘积Schur三元组的性质，探讨了在不同环境下避免单色解a+b=c的着色问题。