Let \(K\) be a field, \(A_{n}=(x_{ij})\) an \(n \times n\) generic skew-symmetric matrix of indeterminates, \(K[x_{ij}]\) the polynomial ring over \(K\), and \(P_{n,r}\) the ideal of Pfaffians of degree \(r\) of \(A_{n}\). J. Herzog and N. V. Trung [Adv. Math. 96, 1–37 (1992; Zbl 0778.13022)] constructed a term order on the monomials of \(K[x_{ij}]\) for which the standard generators of \(P_{n,r}\) constitute a Gröbner basis. In the paper under review the authors determine a different term order leading to the same Gröbner basis, such that the corresponding initial ideal of \(P_{n,r}\) is squarefree and equal to the Stanley-Reisner ideal of a join of a simplicial sphere and a simplex. In addition they study similar term orders for certain determinantal ideals. They prove that the initial ideals are squarefree and also the corresponding simplicial complexes do not decompose into a join of a simplicial sphere and a simplex.