Sanhedrin, Chapter Two, Mishnah Three
Mishnah three discusses the procedures of mourning if a kings family member dies.
1) If any of his near kin die he may not go out of the door of his palace.
a) Rabbi Judah says: If he wishes to follow the bier he may, since we have found that David followed the bier of Avner, as it says, And King David followed the bier (II Samuel 3:31)
b) They answered, That was only to appease the people.
2) When they feed him the funeral meal all the people sit on the floor and he sits on a couch.
This mishnah deals with funeral procedure when one of the kings near relatives dies. Although there is no prohibition in the Torah against the king participating in funerals, the Sages did not think it respectful for a king to be seen in a state of mourning. Rabbi Judah again finds biblical precedent for the king to participate in the funeral (as he found precedent for the king to marry another kings widow in mishnah two). When during the battles between the House of Saul and the House of David, Avner, Sauls army commander, was murdered by Joab, the commander of Davids army for having killed Joabs brother, David goes out after Avners bier. According to Rabbi Judah this is precedent for any king. The Sages reply to Rabbi Judah that David only did so to appease the people so that they wouldnt say that Joab killed Avner with Davids permission. In other, normal, cases it is forbidden for the king to participate in the funeral procession.
When the people feed the king his funeral meal, they sit on the floor, to show their participation in his grief and he sits on a couch, since he is not allowed to truly mourn. Note that the high priest was allowed to sit on a low stool. The Sages were less concerned about the respect shown for the high priest than they were for the respect shown towards the king.