def statistic(pi):
    return len(split(pi))

def split_product(pi, sigma):
    """
    An associative product.

    sage: split_product([[1,2]],[[1,3],[2]])
    {{1, 2, 3, 5}, {4}}

    sage: split_product([[1,3],[2]],[[1,2]])
    {{1, 3, 4, 5}, {2}}
    """
    pi = sorted([sorted(B) for B in pi])
    sigma = sorted([sorted(B) for B in sigma])
    k = len(pi)
    l = len(sigma)
    m = SetPartition(pi).size()
    if k <= l:
        return SetPartition([B + [c+m for c in C] for B, C in zip(pi, sigma)]
                            + [[c+m for c in C] for C in sigma[k:]])
    else:
        return SetPartition([B + [c+m for c in C] for B, C in zip(pi, sigma)]
                            + pi[l:])

@cached_function    
def unsplittable(n):
    """
    sage: [len(unsplittable(n)) for n in range(1,8)]
    [1, 1, 2, 6, 22, 92, 426]
    sage: oeis(_)
    0: A074664: Number of algebraically independent elements of degree n in the algebra of symmetric polynomials in noncommuting variables.

    """
    return Set(SetPartitions(n)).difference(Set([split_product(pi, sigma) for k in range(1,n) for pi in SetPartitions(k) for sigma in SetPartitions(n-k)]))

def split(pi):
    pi = SetPartition(pi)
    n = pi.size()
    if pi in unsplittable(n):
        return [pi]
    for l in range(1, n):
        for sigma in unsplittable(l):
            for kappa in SetPartitions(n-l):
                if pi == split_product(sigma, kappa):
                    return [sigma] + split(kappa)