We consider the Specht module graphs for partitions of the form
and
determine the parity of the number of paths on these directed graphs. For values of
the partition is Lucas perfect, so the associated Specht module will have a
one-dimensional summand exactly when the number of paths on the graph is odd. By
establishing equivalence classes on the set of paths and comparing to smaller
partitions, we are able to demonstrate that none of these Specht modules have a
one-dimensional summand.
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