Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The study of core partitions has been very active in recent years, with the study of  $(s,t)$-cores  — partitions which are both $s$- and $t$-cores %mdash; playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang.  In the present paper,  develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525—1539], studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is ighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this ighted average gives the expected size of the $t$-core of a random $s$-core.
['Matthew Fayers']
I am writing a research paper on a idea $(s,t)$-cores: a weighted version of Armstrong's conjecture
[{"from": "human", "value": "I am writing a research paper on a idea $(s,t)$-cores: a weighted version of Armstrong's conjecture"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The study of core partitions has been very active in recent years, with the study of  $(s,t)$-cores  \u2014 partitions which are both $s$- and $t$-cores %mdash; playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang.  In the present paper,  develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525\u20141539], studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is ighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this ighted average gives the expected size of the $t$-core of a random $s$-core."}]