def to_poset(elt, with_leaves=False, root_to_leaf=False):
    if (not with_leaves) and (not elt):
        # this case needs extra care :(
        from sage.combinat.posets.posets import Poset
        return Poset({})
    return elt.as_ordered_tree(with_leaves).to_poset(root_to_leaf)

@cached_function
def preimages(level):
    result = dict()
    for el in BinaryTrees(level):
        image = to_poset(el)._hasse_diagram.canonical_label().copy(immutable=True)
        result[image] = result.get(image, 0) + 1
    return result

def statistic(x):
    return preimages(x.cardinality()).get(x._hasse_diagram.canonical_label().copy(immutable=True), 0)