A tightness property of relatively smooth permutations

It is well known that many geometric properties of Schubert varieties of type $A$ can be interpreted combinatorially. Given two permutations $w,x\in S_n$ we give a combinatorial consequence of the property that the smooth locus of the Schubert variety $X_w$ contains the Schubert cell $Y_x$. This is a necessary ingredient for the interpretation of recent representation-theoretic results of the author with M'inguez in terms of identities of Kazhdan–Lusztig polynomials.