From out of the city's dark outdoors came 29 rugged men in coonskin caps, moccasins and deerskin suits, trooping into the lobby of the Harrington Hotel Thursday night, carrying knives, powder horns and long-barreled rifles.

"Ma'am, we've been on Amtrak for four days from Flagstaff, Ariz.," the bearded leader of the group told a bewildered desk clerk as bellhops and street toughs looked on in shock. "We're here to claim our rooms."

With that, Arizona's Bill Williams Mountain Men, participants in President-elect Ronald Reagan's inaugural parade Tuesday, announced their arrival in Washington.

Inauguration madness has begun.

For the next three days, the nation's capital will be invaded by big-name Hollywood entertainers, high school and military bands, hundreds of demonstrators, thousands of rejoicing Republicans and a rainbow of Americana -- the Mountain Men, a team of Alaskan husky dog-sled handlers, Indians such as Iron Eyes Cody and others.

Most hotels inside the beltway are booked for the inauguration, which officially begins today. But officials said there are about 1,000 rooms which have not been paid for and may be 'unreserved' this weekend. Most Georgetown restaurants and discos also are booked, and the area's three public airports are gearing up for two or three times as much business as usual.

What the party-goers will find when they arrive are sidewalk hucksters peddling button portraits of Reagan and John Wayne and $1 bumperstickers announcing that Dixon, Ill., the president-elect's birthplace, is the "Home of Ronald Reagan."

They will also find the Reagan inauguration, billed as the best, the biggest and the classiest -- not to mention the most expensive -- in history has its share of contrasts and problems.

And they will also find the Mountain Men, a breed of Americans seemingly as different from the high-class, California tone of the Reagan Inaugural as the president-elect is from Jimmy Carter, the Georgia peanut farmer.

They come from Williams, Ariz., a town of 2,700 people, perched 7,000 feet up the side of a mountain at the gateway to the Grand Canyon. They are plumbers, policemen, linemen, heavy equipment operators, truckers, carpenters and other tradesmen.

They hunt, fish and camp together, sponsor rodeos and steak cookouts, march in parades around Arizona and other western states, and every year pack up their mules and hop on their horses for a 180-mile ride, downhill, to Phoenix for an annual rodeo put on by the Phoenix Jaycees.

The namesake of the 27-year-old club was a 19th century preacher - turned recluse, William Sherley Williams, who trapped along the tributaries of the Gila River in northern Arizona. He later headed a band of mountain men during the 1840s and was a guide and interpreter for many expeditions.

"This is what you call a private club," said troop leader Bill Gano, as he peeled off his deerskin suit in a Harrington Hotel room. "The purpose is to keep Bill Williams' spirit alive, you know, the idea of Old West individualism.

"We're a goodly slice of Americana."

The group was invited to participate in the parade by the presidential inaugural committee on the recommendation of Arizona Sen. Barry Goldwater. They boarded a train in Arizona Tuesday -- "They gave us a special lounge all to ourselves to keep us away from the other passengers," said one member. "Guess they figured we were kinda rowdy folks" -- and arrived in Washington prepared to defend themselves in a city they heard was rather violent.

"Hey, don't write my name on this but we heard if the (gays) don't get you, the muggers will," said one man, downing a bourbon and water in the Hotel's Pink Elephant Room. "Yeah, we're going to be wearing our getups the whole time we're here. But we decided before we left that we wouldn't leave the hotel in groups of less than six guys. We don't want to stir up any trouble."

The spirit of western individualism epitomized by the mountain men ran up against a knot of Washington red tape as soon as the club members set foot in the hotel. The manager was reluctant to allow the mountain men to keep their rifles and knives in their rooms.

"Look," Gano said at one point. "We came all this way with them and I don't believe the men would like to part with them now."

The manager, after a glance at the weary, bearded crew, finally said, "Okay, have it your way, but I can't guarantee someone won't try to break in to get at them."

"Believe 'em," Gano replied, "we'd feel a lot better with our guns by our side."

In a parade where the theme is "America -- A Great New Beginning," the mountain men represent a piece of past Americana that, according to Dean, is becoming more and more difficult to preserve.

"A lot of the ranchland around Williams is owned by corporations now, for tax write-offs," he said. "They use the land like it's some kind of toy, raising stuff like Arabian horses and show cattle. Every year we go on our trail ride to Phoenix we see more and more fences and roads.

"Don't get the idea we're yokels," said he. "We're pretty much like everywhere else in the country, changing every day. We've got an Hispanic on the city police force, a Hopi Indian state cop and a Chinese camera dealer in town whose been there as long as I can remember."

For those who will watch the mountain men and other groups in the parade and attend other official inaugural activities, yesterday offered the first encounters with a ticket distribution system still not perfect.

At Union Station, where inaugural officials are dispersing tickets from red-white-and-blue booths, a dozen sidewalk bums who usually have the run of the station have been herded to one small corner well out of the path of smartly dressed ticket-buyers who pluck bunches of $100 bills from their pocketbooks.

Inaugural officials and other VIPs held a reception yesterday at the station, where some of America's best-known restaurants -- Chasen's of Los Angeles, Bookbinders of Philadelphia and Anthony's Pier 4 of Boston -- gave away free liquor and gourmet samplings while the police carefully watched the street people huddled outside in urine-stained corners to avoid the biting cold.

For the hundreds of volunteers stationed at ticket booths, yesterday's crowd provided a test run of a system that is expected to put 100,000 invited guests together with 200,000 tickets.

"The demand for tickets has been unprecedented," said Walter Gold, who has been in charge of dispensing tickets for the last three Republican presidents' inaugurals. "The demand is incredible. Whatever problems we are having are because we simply do not have as many tickets as people want."

Gold labeled all Friday's mix-ups "minor problems." But Jim Womack of Phoenix reacted a bit differently when he finally got to the front of a line and was given an empty envelope that should have contained two $100 tickets to Tuesday's inaugural ball at the Kennedy Center -- the most popular ball because Republicans from Reagan's home state of California will be sent there.

Officials soon discovered that an additional 109 envelopes were empty because of mistakes by some of the 3,500 volunteers at inaugural headquarters. But Gold said that problem "is now under control."

The Smithsonian Institution museums, the scene of more than 50 free musical shows during the inaugural, also had problems Friday. At the Museum of Natural History, workmen labored frantically to finish building a stage for this weekend's concerts in the rotunda and Tuesday's inaugural ball there.

"It's been horrible, I've had to move my trucks from entrances six times this afternoon. We've got to try to coordinate with more than eight different companies and some of them are not completing their part of the work on time," said "Happy" Acosta, sound engineer with Associated Sound, a company providing lighting and sound equipment for several Smithsonian functions.

Reagan's four-day inaugural, which officially begins today, is estimated to cost more than $8 million.

While its officials have promised to make this the biggest and best inaugural ever, the public will only be allowed to take part in four free events -- today's opening ceremony and fireworks, and the inaugural parade, a free concert, and more fireworks, all on Tuesday.

Most of the real-show stoppers, such as Monday's star-studded gala at the Capital Centre are by invitation only and tickets can cost as much as $500 per person for some events.