Medal outcomes for six sprinters when ties allowed

Six runners race. Ties are allowed. Gold goes to every runner with nobody ahead of them, silver to every runner with exactly one runner ahead, and bronze to every runner with exactly two runners ahead. Count the number of distinct ways the medals can be handed out.

$873$. Split by the pattern of medals: only golds ($k\ge3$ tied winners), gold then silver (exactly one gold, $\ge2$ silvers), gold then bronze (exactly two golds, $\ge1$ bronze), or one gold, one silver and $\ge1$ bronze. Summing binomial counts over these cases gives $42+156+225+450=873$; a brute-force enumeration of all finishing orders confirms it.