@cached_function

def aux(L):

    if L.cardinality() == 1:

        return 1, 1

    b = L.bottom()

    U = L.upper_covers(b)

    J = set()

    for k in range(1, len(U)+1):

        for S in Subsets(U, k):

            J.add(L.join(S))

    t = L.top()

    C = {}

    for j in J:

        M = L.sublattice(L.interval(j, t))

        M = M.canonical_label()

        l, c = aux(M)

        C[l+1] = C.get(l+1, 0) + c

    l = min(C.keys())

    return l, C[l]

​

def statistic(L):

    return aux(L.canonical_label())[1]

​

​

def parent_initializer(n):

    class MyLattices():

        def __iter__(self):

            if n <= 2:

                for P in Posets(n):

                    if P.is_lattice():

                        yield LatticePoset(P)

            else:

                for P in Posets(n-2):

                    Q = P.with_bounds()

                    if Q.is_lattice():

                        yield LatticePoset(Q)

        def cardinality(self):

            l = [1, 1, 1, 1, 2, 5, 15, 53, 222, 1078, 5994, 37622, 262776, 2018305, 16873364, 152233518, 1471613387, 15150569446, 165269824761, 1901910625578, 23003059864006]

            if n < len(l):

                return l[n]

            else:

                return Infinity

    return MyLattices()

​

element_repr = lambda X: str( ( sorted(X._hasse_diagram.canonical_label().cover_relations()), len(X._hasse_diagram.vertices(sort=False)) ) )

levels = [3, 4, 5, 6, 7]

for level in levels:

    for elt in parent_initializer(level):

        print('%s  =>  %s' % (element_repr(elt), statistic(elt)))