def einsum(
    *operands: OperandType,
    alpha: T = 1.0,  # type: ignore[assignment]
    beta: T = 0.0,  # type: ignore[assignment]
    out: Optional[NDArray[T]] = None,
    contract: Optional[Callable[..., NDArray[T]]] = None,
    optimize: bool = True,
) -> NDArray[T]:
    """Evaluate an Einstein summation convention on the operands.

    Using the Einstein summation convention, many common
    multi-dimensional, linear algebraic array operations can be
    represented in a simple fashion. In *implicit* mode `einsum`
    computes these values.

    In *explicit* mode, `einsum` provides further flexibility to compute
    other array operations that might not be considered classical
    Einstein summation operations, by disabling, or forcing summation
    over specified subscript labels.

    See the `numpy.einsum` documentation for clarification.

    Args:
        operands: Any valid input to `numpy.einsum`.
        alpha: Scaling factor for the contraction.
        beta: Scaling factor for the output.
        out: If provided, the calculation is done into this array.
        contract: The function to use for contraction.
        optimize: If `True`, use the `numpy.einsum_path` to optimize the contraction.

    Returns:
        The calculation based on the Einstein summation convention.
    """
    # Parse the kwargs
    inp, outs, args = _parse_einsum_input(list(operands))  # type: ignore
    subscript = "%s->%s" % (inp, outs)

    # Get the contraction function
    if contract is None:
        contract = {
            "numpy": _contract_numpy,
            "ttdt": _contract_ttdt,
            "tblis": _contract_tblis,
        }[EINSUM_BACKEND.lower()]

    # Perform the contraction
    if not len(args):
        raise ValueError("No input operands")
    elif len(args) == 1:
        # If it's just a transpose, use numpy
        res = _transpose_numpy(subscript, args[0], alpha=alpha, beta=beta, out=out)
    elif len(args) == 2:
        # If it's a single contraction, call the backend directly
        res = contract(subscript, args[0], args[1], alpha=alpha, beta=beta, out=out)
    else:
        # If it's a chain of contractions, use the path optimizer
        args = list(args)
        path_kwargs = dict(optimize=optimize, einsum_call=True)
        contractions = np.einsum_path(subscript, *args, **path_kwargs)[1]
        for contraction in contractions:
            inds, idx_rm, einsum_str, remain = list(contraction[:4])
            contraction_args = [args.pop(x) for x in inds]  # type: ignore
            if alpha != 1.0 or beta != 0.0:
                raise NotImplementedError("Scaling factors not supported for >2 arguments")
            if len(contraction_args) == 1:
                a = contraction_args[0]
                res = _transpose_numpy(
                    einsum_str, a, alpha=types[float](1.0), beta=types[float](0.0), out=None
                )
            else:
                a, b = contraction_args
                res = contract(
                    einsum_str, a, b, alpha=types[float](1.0), beta=types[float](0.0), out=None
                )
            args.append(res)

    return res

def dirsum(*operands: Union[str, tuple[int, ...], NDArray[T]]) -> NDArray[T]:
    """Direct sum of arrays.

    Follows the `numpy.einsum` input conventions.

    Args:
        operands: Any valid input to `numpy.einsum`.

    Returns:
        The direct sum of the arrays.
    """
    # Parse the input
    input_str, output_str, input_arrays = _parse_einsum_input(list(operands))
    input_chars = input_str.split(",")

    for i, (chars, array) in enumerate(zip(input_chars, input_arrays)):
        if len(chars) != array.ndim:
            raise ValueError(f"Dimension mismatch for array {i}.")
        if len(set(chars)) != len(chars):
            unique_chars = "".join(set(chars))
            array = einsum(f"{chars}->{unique_chars}", array)
            input_chars[i] = unique_chars
        if i == 0:
            res = array
        else:
            shape = res.shape + (1,) * array.ndim
            res = res.reshape(shape) + array

    # Reshape the output
    res = einsum(f"{''.join(input_chars)}->{output_str}", res)
    res.flags.writeable = True

    return res