Mayor Marion Barry yesterday ordered the city's fire department to proceed with three overdue promotions after a city investigation found "no hard evidence of cheating or other irregularities" in the administration of a departmental promotion exam given last June.

Two weeks ago, the mayor had imposed a delay of up to one month on scheduled promotions because the union representing the city's firemen, Local 36 of the International Association of Firefighters, had complained of possible irregularities in the testing of 285 firefighters for promotion to sergeant.

Yesterday, Barry released a fourpage report from acting inspector general David Legge, the city government's principal auditor, who said, "not one individual could furnish any direct or specific evidence that would substantiate the allegations of cheating. All of those interviewed who had made a formal complaint to the fire department could state only that their complaint was based on hearsay or rumors and not facts."

On the basis of those findings, Barry ordered three promotions -- one each to captain, lieutenant and sergeant -- to proceed as scheduled as of Feb. 11. "This should put this issue to rest for all time," the mayor said through his press secretary, Florence Tate. "I don't want to hear about all this any more.

"We had a fair and impartial investigation," he said. "It turned up nothing. Promotions will proceed as scheduled." The firefighters involved will receive appropriate pay increases retroactive to Feb. 11, the mayor said.

The controversy over the examination had racial overtones, because the union making the allegations is predominantly white. Sixteen of the top 20 scorers on the sergeants exam were black.

A group representing the black firemen alleged that the union's concerns were motivated by racism because for the first time blacks had scored highest in the biennial exams.

However, the union insisted that it was not alleging that blacks had to cheat in order to make high marks on the test. Rather, union spokesmen said, their concern was with possible irregularities in the administration of the test.

These included allegations that some firefighters were given test questions in advance and other firefighters had exchanged answers during a break in the administration of the test.

It also was alleged that a high ranking fire official who served on a sixmember panel that helped prepare the examination had told some firefighters what material would not be included in the examination.

And, accusers said, a female employe of the then-U.S. Civil Service Commission, who helped prepare the test and had advance knowledge of the examination questions, may have provided information on the examination through a personal relationship she had with a member of the fire department.

Legge said yesterday that he interviewed about 20 to 25 members of the fire department, including those who made the alegations and those who were accused. None of those interviewed, however, gave statements under oath, Legge said.

In general, Legge's report found that none of those accused would say that they were guilty of the actions they were supposed to have committed. In investigations conducted by other city personnel, some of those accused had refused to answer questions and in one instance, sources said, had not volunteered to take a lie detector test.

Legge said yesterday that no one had refused to answer any questions his investigators asked.

Fire Chief Jefferson W. Lewis said last night, "I'm in agreement with what the mayor is doing. They didn't find anything. The promotions should go ahead."

William Hoyle, president of the firemen's union, said he had not seen the report but his organization would accept Legge's recommendations if it found that the investigators had reasonable logic to explain the alleged irregularities.

Theodore Holmes, president of the black group, the Progressive Firefighters Association, said he was "very happy" and "very pleased" by the results of the investigation and by Barry's recommendation.