def METHOD_NAME(self):
    r"""
    The hook lengths of the elements of the d-complete poset.
    See [KY2019]_ for the definition of hook lengths for d-complete posets.
    TESTS::
        sage: from sage.combinat.posets.d_complete import DCompletePoset
        sage: P = DCompletePoset(DiGraph({0: [1, 2], 1: [3], 2: [3], 3: []}))
        sage: P._hooks
        {0: 1, 1: 2, 2: 2, 3: 3}
        sage: from sage.combinat.posets.poset_examples import Posets
        sage: P = DCompletePoset(Posets.YoungDiagramPoset(Partition([3,2,1]))._hasse_diagram.reverse())     # optional - sage.combinat
        sage: P._hooks                                                                                      # optional - sage.combinat
        {0: 5, 1: 3, 2: 1, 3: 3, 4: 1, 5: 1}
    """
    hooks = {}
    min_diamond = {}  # Maps max of double-tailed diamond to min of double-tailed diamond
    max_diamond = {}  # Maps min of double-tailed diamond to max of double-tailed diamond
    H = self._hasse_diagram
    diamonds, _ = H.diamonds()  # Tuples of four elements that are diamonds
    diamond_index = {}  # Map max elmt of double tailed diamond to index of diamond
    # Find all the double-tailed diamonds and map the mins and maxes
    for index, d in enumerate(diamonds):
        min_diamond[d[3]] = d[0]
        max_diamond[d[0]] = d[3]
        diamond_index[d[3]] = index
        min_elmt = d[0]
        max_elmt = d[3]
        while True:
            potential_min = H.neighbors_in(min_elmt)
            potential_max = H.neighbors_out(max_elmt)
            # Check if any of these make a longer double tailed diamond
            found_diamond = False
            for (mn, mx) in [(i, j) for i in potential_min for j in potential_max]:
                if len(H.neighbors_in(mx)) != 1:
                    continue
                if len(H.all_paths(mn, mx)) == 2:
                    # Success
                    min_elmt = mn
                    max_elmt = mx
                    min_diamond[mx] = mn
                    max_diamond[mn] = mx
                    diamond_index[mx] = index
                    found_diamond = True
                    break
            if not found_diamond:
                break
    # Compute the hooks
    queue = deque(H.sources())
    enqueued = set()
    while queue:
        elmt = queue.popleft()
        if elmt not in diamond_index:
            hooks[elmt] = H.order_ideal_cardinality([elmt])
        else:
            diamond = diamonds[diamond_index[elmt]]
            side1 = diamond[1]
            side2 = diamond[2]
            hooks[elmt] = hooks[side1] + hooks[side2] - hooks[min_diamond[elmt]]
        enqueued.add(elmt)
        for c in H.neighbors_out(elmt):
            if c not in enqueued:
                queue.append(c)
                enqueued.add(c)
    poset_hooks = {self._vertex_to_element(key): ZZ(value) for (key, value) in hooks.items()}
    return poset_hooks