Explicit dimension formula for modular simple S_n-modules D_λ

Determine an explicit, closed-form formula for the dimension of the irreducible S_n-module D_λ over a field of characteristic p > 0, where D_λ is the unique simple quotient of the Specht module S_λ corresponding to a p-regular partition λ ⊢ n.

Background

In the modular representation theory of the symmetric group S_n over a field k of characteristic p > 0, the Specht modules S_λ (defined over Z and reduced modulo p) may become reducible. For p-regular partitions λ ⊢ n, the p-modular reduction of S_λ has a unique simple quotient denoted D_λ, and all irreducible S_n-modules over k arise in this way.

Despite the parametrization of simples via p-regular partitions being known, obtaining explicit dimensions of these simple modules is difficult. The paper explicitly notes that an explicit formula for dim D_λ is still unknown, highlighting a concrete and longstanding open problem in the field.