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 `windowing' property holds for all such Jack L-R coefficients. Furthermore, we propose an extension of the `Factorization' property for Schur L-R due to King-Tollu-Toumazet to the Jack case. These properties provide a vast set of relations between the Jack L-R coefficients and allow for their direct computation in a certain large class of cases.