Partial symmetries of iterated plethysms

We study the coe cients of the Schur expansion of iterated plethysms, sλ1◦sλ2◦· · · sλk = ∑ aμsμ . For certain choices of λ1, ..., λk , these coe cients express partial symmetries. We formalize this symmetry by means of an involution on partitions, and show that if f has this symmetry, then sλ ◦ f too for λ ∈ Par(2). We give explicit formulas for the plethysms s2 ◦ sb ◦ sa and sc ◦ s2 ◦ sa , a, b, c ∈ N≥2 and show that they have this partial symmetry. Finally, we make some remarks and conjectures regarding unimodality and asymptotic normality of the coe cients involved.