A $q$-multinomial expansion of LLT coefficients and plethysm multiplicities

Lascoux, Leclerc and Thibon\cite{LLT} introduced a family of symmetric polynomials, called LLT polynomials. We prove a $q$-multinomial expansion of the coefficients of LLT polynomials in the case where $ \boldsymbol{\mu} = \underbrace{(\mu,…,\mu)}_{n}$anddefinea$q$-analog of a sum of the plethysm multiplicities.