The Stanley Conjecture Revisited

In the seminal work of Stanley, several conjectures were made on the structure of Littlewood-Richardson coefficients for the multiplication of Jack symmetric functions. Motivated by recent results of Alexandersson and the present author, we postulate that a certain 'windowing' property holds for all such Jack L-R coefficients. This property provides a vast set of relations between these coefficients and allows for their direct computation in certain novel cases. We demonstrate compatibility between our windowing conjecture and the conjectures of Stanley, with the hope of illuminating the structures within.