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Permutations and associated algebras allow the construction of half andquarter BPS operators in maximally supersymmetric Yang Mills theory with$U(N)$, $SO(N)$ and $Sp(N) $ gauge groups. The construction leads to bases forthe operators, labelled by Young diagrams and associated group theory data,which have been shown to be orthogonal under the inner product defined by thefree field two-point functions. In this paper, we study in detail theorientifold projection map between the Young diagram basis for $U(N)$ theoriesand the Young diagram basis for $ SO(N)$ (and $ Sp(N)$) half-BPS operators. Wefind a simple connection between this map and the plethystic refinement of theLittlewood Richardson coefficients which couple triples of Young diagrams wheretwo of them are identical. This plethystic refinement is known to be computableusing an algorithm based on domino tilings of Young diagrams. We discuss thedomino combinatorics of the orientifold projection map in terms of giantgraviton branes. The permutation construction of $SO(N)$ operators is used tofind large N generating functions. The structure of these generating functionsis elucidated using the combinatorics of words, organised according to theirperiodicity. Aperiodic words in the $U(N)$ theory are Lyndon words and an$SO(N)$ analogue of Lyndon words is defined using a minimal periodicitycondition. We calculate the normalization factor for the orthogonal basis ofYoung diagram operators in the baryonic sector of $SO(N)$.

BPS operators in $\mathcal{N}=4$ $SO(N)$ super Yang-Mills theory: plethysms, dominoes and words