
    In a hospital emergency room triage system with n distinct severity levels, develop an algorithm to determine the number of possible ways to assign priorities to incoming patients such that exactly m severity levels have a higher priority patient immediately following a lower priority patient. This calculation is crucial for validating and optimizing the triage system's patient prioritization patterns.
    

    
def eulerian_num(n, m):
    """
    In a hospital emergency room triage system with n distinct severity levels, develop an algorithm to determine the number of possible ways to assign priorities to incoming patients such that exactly m severity levels have a higher priority patient immediately following a lower priority patient. This calculation is crucial for validating and optimizing the triage system's patient prioritization patterns.
    

    """
def eulerian_num(n, m):
    """
    In a hospital emergency room triage system with n distinct severity levels, develop an algorithm to determine the number of possible ways to assign priorities to incoming patients such that exactly m severity levels have a higher priority patient immediately following a lower priority patient. This calculation is crucial for validating and optimizing the triage system's patient prioritization patterns.
    

    """
    if m >= n or n == 0:
        return 0
    if m == 0:
        return 1
    return (n - m) * eulerian_num(n - 1, m - 1) + (m + 1) * eulerian_num(n - 1, m)