
    Calculate the number of unique ways a student can progress through n learning modules while experiencing exactly m performance fluctuations, where a fluctuation occurs when a student performs better in a later module compared to an earlier one. The result helps educators understand the statistical likelihood of different learning progression patterns to better personalize curriculum pacing and support.
    

    
def eulerian_num(n, m):
    """
    Calculate the number of unique ways a student can progress through n learning modules while experiencing exactly m performance fluctuations, where a fluctuation occurs when a student performs better in a later module compared to an earlier one. The result helps educators understand the statistical likelihood of different learning progression patterns to better personalize curriculum pacing and support.
    

    """
def eulerian_num(n, m):
    """
    Calculate the number of unique ways a student can progress through n learning modules while experiencing exactly m performance fluctuations, where a fluctuation occurs when a student performs better in a later module compared to an earlier one. The result helps educators understand the statistical likelihood of different learning progression patterns to better personalize curriculum pacing and support.
    

    """
    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)