def i_less(a, b, i):
    """
    Return whether a < b for i < i+1 < ... < n < 1 < ... < i-1.
    """
    if min(a, b) >= i or max(a, b) < i:
        return a < b
    return a >= i > b

def Grassmann_necklace(pi):
    """
    Return the Grassmann necklace corresponding to pi.

    sage: pi = Permutation([3,4,2,1])
    sage: Grassmann_necklace(pi)
    [{1, 2}, {2, 4}, {3, 4}, {1, 4}]
    """
    pi = Permutation(pi)
    N = []
    n = len(pi)
    for i in range(1,n+1):
        iWEX = [j for j in range(1,n+1) if j == pi(j) or i_less(j, pi(j), i)]
        N.append(set(iWEX))
    return N

def Gale_less(S, T, i):
    from functools import cmp_to_key
    key = cmp_to_key(lambda a,b: 0 if a == b else int(-1) if i_less(a,b,i) else int(1))
    S_sorted = sorted(S, key=key)
    T_sorted = sorted(T, key=key)
    return all(s == t or i_less(s, t, i) for s,t in zip(S_sorted, T_sorted))

def positroid(N):
    """
    Return the positroid corresponding to the Grassmann necklace.

    sage: pi = Permutation([3,4,2,1])
    sage: N = Grassmann_necklace(pi)
    sage: positroid(N)
    [{1, 2}, {1, 3}, {1, 4}, {2, 4}, {3, 4}]

    sage: [set([len(positroid(Grassmann_necklace(p))) for p in orbit(pi)]) for pi in Permutations(4)]


    """
    n = len(N)
    d = len(N[0])
    result = []
    for B in Subsets(range(1,n+1), d):
        if all(B == N[j] or Gale_less(N[j], B, j+1) for j in range(n)):
            result.append(B)

    return result

def statistic(pi):
    return len(positroid(Grassmann_necklace(pi)))