Restricted Stirling and Lah number matrices and their inverses

If we have 1,2∈R and if for all n∈R with n odd and n≥3, we have n±1∈R, we additionally show that each entry of [{nk}R]n,k≥1−1, [[nk]R]−1n,k≥1 and [L(n,k)R]n,k≥1−1 is up to an explicit sign the cardinality of a single explicitly defined family of labeled forests. With R as before we also do the same for restriction sets of the form R(d)={d(r−1)+1:r∈R} for all d≥1. Our results also provide combinatorial interpretations of the kth Whitney numbers of the first and second kinds of Πn1,d, the poset of partitions of [n] that have each part size congruent to 1 mod d.