Perming hair the day after dying it is not recommended because it can cause damage according to about.com. The rule of thumb is to dye following a perm.
Dying permed hair is preferred because the perm opens the cuticle layers, leading to a richer and longer-lasting color. However, getting a perm right after dying the hair could cause severe damage.
If the hair was recently dyed, wait at least 10 days before having it permed to avoid damage. Do not wait too long, or else managing the right time between perming and dying could become difficult and confusing. This same advice is followed when relaxing the hair. Relaxing uses chemicals to remove natural curls.