The Plethysm $s_\lambda[s_\mu]$ at Hook and Near-Hook Shapes

We completely characterize the appearance of Schur functions corresponding to partitions of the form $\nu = (1^a, b)$ (hook shapes) in the Schur function expansion of the plethysm of two Schur functions, $$s_\lambda[s_\mu] = \sum_{\nu} a_{\lambda, \mu, \nu} s_\nu.$$ Specifically, we show that no Schur functions corresponding to hook shapes occur unless $\lambda$ and $\mu$ are both hook shapes and give a new proof of a result of Carbonara, Remmel and Yang that a single hook shape occurs in the expansion of the plethysm $s_{(1^c, d)}[s_{(1^a, b)}]$. We also consider the problem of adding a row or column so that $\nu$ is of the form $(1^a,b,c)$ or $(1^a, 2^b, c)$. This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case.