If you are referring to their number in the race, then yes. Obviously, their number will vary from race to race as they have different post positions for different races.
It's either randomly drawn, or the names are randomly drawn and the trainers get to pick what position they want in the order their horse's name is drawn.
In Thoroughbred racing, the numbers are determined by starting position, with number 1 having the pole position (closest to the inside) and the largest number being the furthest to the outside.
Call To The Post.
The Call to The Post
Post position draw is on the Wednesday before the race.
The bugler blows "Call to Post"
Call to Post
It means the first person or animal to pass the finish line in a race. Its generally referring to a horse in a horse race though.
In horse racing the odds are determined by the bets placed. Betting stops when the race starts. These determine the final odds.
The Call To The Post, also known as First Call