None, at least that is the idea advanced by John Dickie in his book Cosa Nostra: A history of the Sicilian Mafia

There is an old tale that goes like this:

It is said that on the morning of 14 July 1943, an American fighter plane flew low over Villalba. It naturally drew the people out into the streets. When it roared down to rooftop level they could see, attached to its fuselage, a golden-yellow flag with a large L in the centre. As it passed over the house belonging to the parish priest, Monsignor Giovanni Vizzini, the pilot dropped a small package. But it was intercepted by an Italian soldier and passed on to the commander of the local carabinieri.

This takes place four days after the beginning of Operation Husky. The story continues:

Yet the fighter plane returned the following day, still bearing its unusual banner. Another package was dropped, and this time it found its way to the right person. Its nylon wrapping bore the Sicilian words ‘zu Calò’—’Uncle Calò’—who was mafia boss Don Calogero Vizzini, the priest’s older brother. It was picked up by the Vizzinis’ butler who took it to his master. It was found to contain a golden-yellow silk handkerchief with a large black L in the centre. That very evening, the story goes, a rider left Villalba with a message for a certain ‘zu Peppi’ in Mussomeli. The message read as follows: ‘On Tuesday 20th Turi will leave for the fair at Cerda with the calves. I will set off the same day with the cows, the oxen, and the bull. Prepare the kindling for the fruit and organize pens for the animals. Tell the other overseers to get ready.’

The letter was written in secret code, according to urban legend the code words imply that Mafia members are to meet with Allied soldiers and provide aid (the codes are farm animals and farm buildings, so soldiers are reffered to as "cows", tanks as "oxens", "pens" refers to aid, the "bull" is the commander of the allied unit)

The story keeps going:

On the afternoon of 20 July, three tanks duly rumbled up to the gates of Villalba. The turret of the first bore the same yellow flag with the large L in the middle. An American officer appeared from the hatch. In a Sicilian accent slurred by years in the States, he respectfully asked for Don Calò. Word reached the old capomafioso at home. He was four days away from his sixty-sixth birthday. On hearing of the Americans’ arrival, he shambled slowly across town in shirtsleeves and tortoiseshell sunglasses, his braces straining to keep a pair of crumpled trousers tethered high over the improbable protrusion of his gut. When he reached the Americans, he wordlessly proffered the silk hankerchief his butler had picked up. Along with his nephew— who spoke English because he had not long returned from the States—he then climbed up on to the tank and was driven away.

The idea was that Luck Luciano had arranged the meeting and set up contact between the allied soldiers and the Mafia. Hence the big "L" on the flag that was dropped.

Finally the story ends:

Six days later Don Calò returned to Villalba in a big American car, his mission accomplished. A perfectly executed pincer movement had brought the calves, cows, and oxen together at Cerda, thus completing the Allied conquest of central Sicily. Now Don Calò, with his American backers, was ready to return the mafia to its rightful place in Sicilian society after the dark days of Fascism.

Now the truth is this story is pretty apocryphal; what most likely happened was that the Don simply led a procession of people to greet the allied soldiers and that got morphed into the urban legend I just recounted.

Now a lot of Mafia backed administrators (and in many cases the Mobsters themselves) took power after the allied advance through Sicily, but that isn't evidence of collaboration. It was more due to the fact that the allies often didn't have pay attention to who they installed as mayors in the various towns they took over; most often the allies took the advice of the first person they could find who could speak English. So naturally the mob took advantage and people looked to the established mobsters to provide leadership in the post Fascist upheaval.

There was some American collaboration with the mob on the homefront. See in early 1942 an America liner caught fire, and sank in New York harbor. It was an accident, but at the time people assumed it was sabotage. To prevent further "acts of sabotage" the Americans sought out local mobsters who were in control of the waterfront. They contacted local mobster Joseph "Socks" Lanza who controlled a fishing business on the waterfront as well as being a captain in a local crime family. He provided false documents and protection to navy agents who covertly investigated dockworkers. On Lanza's recommendation, the American government interviewed Luciano on the subject of potential German spies on the waterfront.

That was the full extent of Luciano's collaboration, there is no evidence that he arranged contacts between the Sicilian mob and the allies, or that he worked out some sort of deal for his freedom (he was in prison at the time on prostitution related charges). But it still continues to be a popular legend in Sicily to this day.