def increment_first_part(p,length=None):
    '''
        Increment the part with the smallest index `i` > 1 in partition `p` when such `i` exists.
        When `length` is given it bound the length of the resulting partition. Otherwise, it may become longer than `p`.
    '''
    q = list(p) + [0]*(0 if length is None else length-len(p))
    try:
        # find smallest index i>0 such that q[i] can be incremented
        i = next(i for i in range(1,len(q)) if q[i]<q[i-1])
        q[i] += 1
    except StopIteration:
        if length is None:
            # grow length of the partition
            q.append(1)
    return Partition(q)

def mapping(p):
    if len(p)==0 or p[0]==1: return p
    m0 = max(i+1 for i in range(len(p)) if p[i]==p[0])   # multiplicity of part p[0]
    q = list(p)
    for i in range(m0): q[i] -= 1
    for i in range(m0): q = increment_first_part(q)
    return Partition(q)