From Longman Dictionary of Contemporary Englishbutton-downˈbutton-down adjective [only before noun] 1 DCa button-down shirt or collar has the ends of the collar fastened to the shirt with buttons2 a button-down company or style is formal and traditional He didn’t fit in with the button-down culture of his new boss.
Examples from the Corpus
button-down• Crew necks were worn with button-down collars kept inside the neck.• He was wearing an old raincoat over his usual gray suit, button-down shirt, and necktie.• White bucks, chinos, a button-down shirt open at the collar.• Watch too for button-down shirts on a Ralph Lauren tip at £69.• Like Southfork Ranch, the seating is in crusty burgundy leatherette button-down style.• Daisy ring-pulls dangle from button-down white sun panels.From Longman Business Dictionarybutton-downˈbutton-down (also buttoned-down) adjective [only before a noun] a button-down company or style is formal and traditionalWearing cowboy boots and open-necked shirts, he did not fit in with the button-down culture of IBM. 