This song is "The Blues done come back" by Count Basie Orchestra.
Is this style swing, blues or jazz? As the link describes, this song is from 1953. Was swing still popular during the 50s?
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This song is swing music, a subdivision of jazz. Swing is also often referred to as "big-band music". Swing was at the height of its popularity in the 1930's and 40's. It was already declining in popularity by 1953. This music would have seemed slightly old-fashioned to the musicians that were pioneering newer styles in 1953 (bebop, rhythm and blues, electric blues).
It is not blues music as that genre is most commonly understood. The use of the word "blues" in the title is not unusual in that many records from the jazz era used the term "blues" broadly.