  We introduce a class of posets, which includes both ribbon posets (skew
shapes) and $d$-complete posets, such that their number of linear extensions is
given by a determinant of a matrix whose entries are products of hook lengths.
We also give $q$-analogues of this determinantal formula in terms of the major
index and inversion statistics. As applications, we give families of tree
posets whose numbers of linear extensions are given by generalizations of Euler
numbers, we draw relations to Naruse-Okada's positive formulas for the number
of linear extensions of skew $d$-complete posets, and we give polynomiality
results analogous to those of descent polynomials by Diaz-L\'opez, Harris,
Insko, Omar, and Sagan.
