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