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Identifier

St001614: Skew partitions ⟶ ℤ

Values

[[1],[]] => 1

[[2],[]] => 2

[[1,1],[]] => 1

[[2,1],[1]] => 2

[[3],[]] => 3

[[2,1],[]] => 1

[[3,1],[1]] => 3

[[2,2],[1]] => 1

[[3,2],[2]] => 3

[[1,1,1],[]] => 3

[[2,2,1],[1,1]] => 3

[[2,1,1],[1]] => 3

[[3,2,1],[2,1]] => 3

[[4],[]] => 4

[[3,1],[]] => 1

[[4,1],[1]] => 4

[[2,2],[]] => 4

[[3,2],[1]] => 2

[[4,2],[2]] => 4

[[2,1,1],[]] => 1

[[3,2,1],[1,1]] => 4

[[3,1,1],[1]] => 1

[[4,2,1],[2,1]] => 4

[[3,3],[2]] => 1

[[4,3],[3]] => 4

[[2,2,1],[1]] => 4

[[3,3,1],[2,1]] => 4

[[3,2,1],[2]] => 4

[[4,3,1],[3,1]] => 4

[[2,2,2],[1,1]] => 1

[[3,3,2],[2,2]] => 1

[[3,2,2],[2,1]] => 4

[[4,3,2],[3,2]] => 4

[[1,1,1,1],[]] => 2

[[2,2,2,1],[1,1,1]] => 4

[[2,2,1,1],[1,1]] => 4

[[3,3,2,1],[2,2,1]] => 4

[[2,1,1,1],[1]] => 4

[[3,2,2,1],[2,1,1]] => 4

[[3,2,1,1],[2,1]] => 4

[[4,3,2,1],[3,2,1]] => 4

[[5],[]] => 5

[[4,1],[]] => 1

[[5,1],[1]] => 5

[[3,2],[]] => 5

[[4,2],[1]] => 1

[[5,2],[2]] => 5

[[3,1,1],[]] => 5

[[4,2,1],[1,1]] => 5

[[4,1,1],[1]] => 5

[[5,2,1],[2,1]] => 5

[[3,3],[1]] => 5

[[4,3],[2]] => 1

[[5,3],[3]] => 5

[[2,2,1],[]] => 5

[[3,3,1],[1,1]] => 5

[[3,2,1],[1]] => 5

[[4,3,1],[2,1]] => 5

[[4,2,1],[2]] => 5

[[5,3,1],[3,1]] => 5

[[3,2,2],[1,1]] => 5

[[4,3,2],[2,2]] => 5

[[4,2,2],[2,1]] => 5

[[5,3,2],[3,2]] => 5

[[2,1,1,1],[]] => 1

[[3,2,2,1],[1,1,1]] => 5

[[3,2,1,1],[1,1]] => 5

[[4,3,2,1],[2,2,1]] => 5

[[3,1,1,1],[1]] => 5

[[4,2,2,1],[2,1,1]] => 5

[[4,2,1,1],[2,1]] => 5

[[5,3,2,1],[3,2,1]] => 5

[[4,4],[3]] => 1

[[5,4],[4]] => 5

[[3,3,1],[2]] => 5

[[4,4,1],[3,1]] => 5

[[4,3,1],[3]] => 5

[[5,4,1],[4,1]] => 5

[[2,2,2],[1]] => 5

[[3,3,2],[2,1]] => 5

[[4,4,2],[3,2]] => 5

[[3,2,2],[2]] => 5

[[4,3,2],[3,1]] => 5

[[5,4,2],[4,2]] => 5

[[2,2,1,1],[1]] => 1

[[3,3,2,1],[2,1,1]] => 5

[[3,3,1,1],[2,1]] => 5

[[4,4,2,1],[3,2,1]] => 5

[[3,2,1,1],[2]] => 5

[[4,3,2,1],[3,1,1]] => 5

[[4,3,1,1],[3,1]] => 5

[[5,4,2,1],[4,2,1]] => 5

[[3,3,3],[2,2]] => 5

[[4,4,3],[3,3]] => 5

[[4,3,3],[3,2]] => 5

[[5,4,3],[4,3]] => 5

[[2,2,2,1],[1,1]] => 1

[[3,3,3,1],[2,2,1]] => 5

[[3,3,2,1],[2,2]] => 5

[[4,4,3,1],[3,3,1]] => 5

[[3,2,2,1],[2,1]] => 5

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Description

The cyclic permutation representation number of a skew partition.
This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.
See St001527The cyclic permutation representation number of an integer partition. for the restriction of this statistic to integer partitions.

References

[1] Alexandersson, P., Amini, N. The cone of cyclic sieving phenomena MathSciNet:3922156

Code

def fdeg(p):
    R. = QQ[]
    f = prod([ 1-q*q^j for j in range(p.size()) ])
    b = sum([ j*k for j, k in enumerate(p) ])
    return (q^b)*(f/p.hook_polynomial(q,q))

def fake_degree(a):
    R. = QQ[]
    if a == 0:
        return R(0)
    else:
        z = SymmetricFunctions(QQ).schur()(a)
        return R(sum([z.coefficient(p)*fdeg(p) for p in z.support()]))

def standard_CSP(p, r):
    P = parent(p)
    q = P.gen()
    PP. = PolynomialRing(QQ)
    R. = PP.quo(q^r-1)
    return R(p).lift()

def orbit_lengths(p, r):
    p = standard_CSP(p, r)
    p = {k: p[k] for k in range(r)}
    divs = sorted(divisors(r))
    O = dict()
    for d in divs:
        count = p[r-d]
        if count:
            O[r//d] = count
        if count:
            for k in range(0, r, d):
                p[k] -= count
    assert all(c == 0 for c in p.values()), "%s" % p
    return O

def has_cyclic_permutation_representation(p, r):
    if p == 0:
        return True
    try:
        lo = orbit_lengths(p, r)
    except AssertionError:
        return False
    return all(v in ZZ and ZZ(v) >= 0 for v in lo.values())

def statistic(la):
    s = SymmetricFunctions(QQ).schur()
    f = fake_degree(s(la))
    for r in reversed(divisors(la.size())):
        if has_cyclic_permutation_representation(f, r):
            return r

# alternative implementation using Theorem 2.7
def has_cyclic_permutation_representation_slow(p, r):
    for k in divisors(r):
        if sum(moebius(k//j) * p.subs(q=QQbar.zeta(r)^j) for j in divisors(k)) < 0:
            return False
    return True

def statistic_slow(la):
    s = SymmetricFunctions(QQ).schur()
    f = fake_degree(s(la))
    for r in reversed(divisors(la.size())):
        if has_cyclic_permutation_representation_slow(f, r):
            return r

Created

Sep 28, 2020 at 11:17 by Martin Rubey

Updated

Sep 28, 2020 at 11:17 by Martin Rubey