-- PHOTO MOVED IN ADVANCE AND NOT FOR USE - ONLINE OR IN PRINT - BEFORE OCT. 14, 2018. -- In a handout photo from VisitLEX, Lexington, Ky., the capital of American horse country and the 15-county Bluegrass Region. Eager to expand audiences for racing, stately horse farms, noted for their architecturally distinct barns and pastures encircled by double-railed fences, are throwing open the barn doors to tours. (Gene Burch/VisitLEX via The New York Times) NO SALES; FOR EDITORIAL USE ONLY WITH NYT STORY SLUGGED KY LEXINGTON ATTRACTIONS BY ELAINE GLUSAC FOR OCT. 14, 2018. ALL OTHER USE PROHIBITED. --