The final meal before the fast of Tish'ah BeAb (Se'uddah Mafseqeth) cannot be eaten sitting on regular chairs or even cushions, but should be eaten sitting on a mat or plank that is less than a Tefah high (about four inches). This assumes, of course, that the person is physically able to do so. According to the Qabbalah, one should not sit directly on the floor.
When three or more men sit together to eat a meal with bread (which requires Birkath Hammazon), they must say Zimmun (the invitation to join in the reading of Birkath Hammazon). Zimmun, indicates a state of permanence. Since we do not want the atmosphere of mourning to be a permanent one, three males should not sit together during this meal, so that they will not become obligated to recite the Zimmun.
(See Ben Ish Hai, Shanah Aleph, Parashath Debarim, Oth Kaf)
