def weighted_digit_sum(n):
    """
    Return the weighted sum of binary digits of ``n`` where ``i``th digit from the right has weight ``i-1``.

    INPUT:

    - ``n`` -- integer.

    OUPUT:

    the weighted digit sum of ``n``.

    EXAMPLE::

        sage: weighted_digit_sum(5)
        2
    """
    return sum([i[0]*eval(i[1]) for i in enumerate(Integer(n).binary()[::-1])])

def number_of_odd_dimensional_partitions(n):
    """
    Return the number of odd dimensional irreducible representations of the ``n``th symmetric group.

    INPUT:

    - ``n`` -- integer.

    OUTPUT:

    - number of partitions for which the associated irreducible representation of the symmetric group
      in ``n`` letters has even dimension.

    EXAMPLE::

        sage: number_of_odd_dimensional_partitions(5):
        4

    ALGORITHM:

    Return ``2**weighted_digit_sum(n)``.
    """
    return 2**weighted_digit_sum(n)

def number_of_chiral_partitions(n, v=None):
    """
    Return the number of partitions ``la`` of ``n`` for which `f_{\lambda/(1,1)}` is odd

    INPUT:

    - ``n`` -- integer.

    OUTPUT:

    The number of partitions of ``n`` for which the corresponding irreducible representation of
    the symmetric group has non-trivial determinat.

    EXAMPLE::

        sage: number_of_chiral_partitions(3)
	2
    """
    def base_case_odd(k, v = v):
        if k == 1:
            if v is None:
                return 2
            elif v == 0:
                return 1
            elif v == 1:
                return 1
            else:
                return 0
        else:
            if v is None:
                return sum([base_case_odd(k, v) for v in range(k+1)])
            if v == 0:
                return 2^(k-1)
            elif v > 0 and v < k:
                return 2^((v+1)*k - 2*(v+1) - v*(v-1)/2)
            elif v == k:
                return 2^(k*(k-1)/2)
            else:
                return 0
    def base_case_even(k, v = v):
        if k == 1:
            if v is None or v == 0:
                return 1
            else:
                return 0
        else:
            if v is None:
                return sum([base_case_even(k, v) for v in range(k)])
            if v == 0:
                return 2^(k-1)
            elif v > 0 and v < k:
                return 2^((v+1)*k - 2*(v+1) - v*(v-1)/2)
            else:
                return 0
    if n == 0 or n == 1:
        return 0
    k = valuation(n, 2)
    if k == 0:
        w = valuation(n - 1, 2)
        m = n - (2^w + 1)
        return number_of_odd_dimensional_partitions(m)*base_case_odd(w, v = v)
    else:
        m = n - 2^k
        return number_of_odd_dimensional_partitions(m)*base_case_even(k, v = v)