  We investigate the connections between patterns in permutations and forbidden
configurations in restricted elections, first discovered by Lackner and
Lackner, in order to enhance the approach initiated by the two mentioned
authors. More specifically, our achievements are essentially two. First, we
define a new type of domain restriction, called enriched group-separable.
Enriched group-separable elections are a subset of group-separable elections,
which describe a special, still natural, situation that can arise in the
context of group-separability. The exact enumeration of group-separable
elections has been very recently determined by Karpov. Here we give a recursive
characterization for enriched group-separable elections, from which we are able
to find a recurrence relation and a closed formula expressing their number. Our
second achievement is a generalization of a result of Lackner and Lackner,
concerning the connection between permutation patterns and forbidden
configurations with 3 voters. Our result relates forbidden configurations with
the strong order on pairs of permutations, a notion which is still largely
undeveloped, and suggests a potential approach for the determination of upper
bounds for restricted elections whose forbidden configurations contains at
least one configuration with 3 voters.
