The quantum K-theory ring QK(X) of a flag variety X encodes the K-theoretic Gromov-Witten invariants of X, defined as arithmetic genera of Gromov-Witten varieties parametrizing curves meeting fixed Schubert varieties. A Pieri formula means a formula for multiplication with a set of generators of QK(X). Such a formula makes it possible to compute efficiently in this ring. I will speak about a Pieri formula for QK(X) when X is a cominuscule Grassmannian, that is, an ordinary Grassmannian, a maximal orthogonal Grassmannian, or a Lagrangian Grassmannian. This formula has a simple statement in terms of order ideals in a partially ordered set that encodes the degree distance between opposite Schubert varieties. This set generalizes both Postnikov's cylinder and Proctor's description of the Bruhat order of X. This is joint work with P.-E. Chaput and N. Perrin.