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Fancy Meeting You Here was conceived as a project for Bing Crosby and Rosemary Clooney, and was recorded in 1958. The arranger/conductor was Billy May, who just so happened to have also done Sinatra's Come Fly with Me (1957), as well as Come Dance with Me! (1959), Swing Along with Me (1961), and Come Swing with Me! (1961).
According to the Wikipedia entry:
In its review, Time magazine called this album, "An infectious musical dialogue between two of the sassiest fancy talkers in the business. C. & C. give slick and witty readings to a selection of retreads — 'On a Slow Boat to China,' 'You Came a Long Way from St. Louis' — and introduce a punchy, potential hit named 'Calcutta.' One of the most intriguing vocal entertainments since Noel Coward had his famous chat with Mary Martin."