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Identifier

Mp00321: Integer partitions —2-conjugate⟶ Integer partitions

Images

[1] => [1]

[2] => [2]

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

[3] => [2,1]

[2,1] => [3]

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

[4] => [2,2]

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

[2,2] => [4]

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

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

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

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

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

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

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

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

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

[6] => [2,2,2]

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

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

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

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

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

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

[2,2,2] => [6]

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

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

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

[7] => [2,2,2,1]

[6,1] => [3,2,2]

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

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

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

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

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

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

[3,2,2] => [6,1]

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

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

[2,2,2,1] => [7]

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

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

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

[8] => [2,2,2,2]

[7,1] => [2,2,2,1,1]

[6,2] => [4,2,2]

[6,1,1] => [4,2,1,1]

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

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

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

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

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

[4,2,2] => [6,2]

[4,2,1,1] => [6,1,1]

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

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

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

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

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

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

[2,2,2,2] => [8]

[2,2,2,1,1] => [7,1]

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

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

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

[9] => [2,2,2,2,1]

[8,1] => [3,2,2,2]

[7,2] => [4,2,2,1]

[7,1,1] => [2,2,2,1,1,1]

[6,3] => [4,3,2]

[6,2,1] => [5,2,2]

[6,1,1,1] => [4,2,1,1,1]

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

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

[5,2,2] => [6,2,1]

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

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

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

[4,3,2] => [6,3]

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

[4,2,2,1] => [7,2]

[4,2,1,1,1] => [6,1,1,1]

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

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

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

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

[3,2,2,2] => [8,1]

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

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

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

[2,2,2,2,1] => [9]

[2,2,2,1,1,1] => [7,1,1]

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

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

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

[10] => [2,2,2,2,2]

[9,1] => [2,2,2,2,1,1]

[8,2] => [4,2,2,2]

[8,1,1] => [4,2,2,1,1]

[7,3] => [3,2,2,2,1]

>>> Load all 1211 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

Return a partition with the same number of odd parts and number of even parts interchanged with the number of cells with zero leg and odd arm length.
This is a special case of an involution that preserves the sequence of non-zero remainders of the parts under division by $s$ and interchanges the number of parts divisible by $s$ and the number of cells with zero leg length and arm length congruent to $s-1$ modulo $s$.
In particular, for $s=1$ the involution is conjugation, hence the name.

Properties

bijective, graded, involutive

Sage code

def reduction(la, s):
    return Partition([p // s for p in la])

def blow_up(nu, s):
    return Partition([s * p for p in nu])

def remainder_sequence(la, s):
    """
    The sequence of non-zero remainders.

    EXAMPLES::

        sage: remainder_sequence(Partition([19,16,14,12,7,5,4,3]), 3)
        (1, 1, 2, 1, 2, 1)
    """
    return tuple(r for p in la if (r := p % s))

def row_sequence(la, s):
    """
    The 0-based indices of the parts with non-zero remainder.

    EXAMPLES::

        sage: row_sequence(Partition([19,16,14,12,7,5,4,3]), 3)
        (0, 1, 2, 4, 5, 6)
    """
    return tuple(i for i, p in enumerate(la) if p % s)

def green_yellow(la, s):
    rho = remainder_sequence(la, s)
    gamma = row_sequence(la, s)
    return tuple((gamma[j], 0) if (j > 0
                                   and rho[j-1] >= rho[j]
                                   and gamma[j] == gamma[j-1] + 1)
                 else (gamma[j], 1)
                 for j in range(len(rho)))

def remainder_diagram(la, s):
    return reduction(la, s), green_yellow(la, s)

def conjugate_remainder_diagram(nu, gy):
    nu_t = nu.conjugate()
    nu = list(nu) + [0]*(gy[-1][0]+1 if gy else 0)
    gy_t = [(nu[j], 1) for j, _ in reversed(gy)]
    return nu_t, gy_t

def from_remainder_diagram(rho, nu, gy, s):
    mu = list(blow_up(nu, s)) + [0]*(gy[-1][0]+1 if gy else 0)
    for i, (j, _) in enumerate(gy):
        mu[j] += rho[i]
    return Partition(mu)

def construction3(rho, nu, gy, verbose=False):
    nu = list(nu) + [0]*(gy[-1][0] if gy else 0)
    gy = list(gy)
    for k in range(len(rho)-1):
        if rho[k] < rho[k+1]:
            continue
        if gy[k+1][1] == 0:
            # yellow
            for j in range(k, -1, -1):
                i = gy[j][0]
                nu[i] += 1
                gy[j+1] = (i + 1, 1)
            gy[0] = (0, 1)
        else:
            # green
            for j in range(k+1):
                i = gy[j][0]
                nu[i] += 1
                gy[j] = (i + 1, 1)

    return Partition(nu), tuple(gy)

def construction3_inverse(rho, nu, gy):
    nu = list(nu)
    gy = list(gy)
    for k in range(len(rho)-2, -1, -1):
        if rho[k] < rho[k+1]:
            continue
        if gy[0][0] == 0:
            # yellow
            for j in range(k+1):
                gy[j] = (gy[j+1][0] - 1, 1)
                nu[gy[j][0]] -= 1
            gy[k+1] = (gy[k+1][0], 0)
        else:
            # green
            for j in range(k+1):
                gy[j] = (gy[j][0] - 1, 1)
                nu[gy[j][0]] -= 1
    return Partition(nu), tuple(gy)

def mapping(la, s=2):
    rho = remainder_sequence(la, s)
    nu, gy = remainder_diagram(la, s)
    nu1, gy1 = construction3(rho, nu, gy)
    nu2, gy2 = conjugate_remainder_diagram(nu1, gy1)
    nu3, gy3 = construction3_inverse(rho, nu2, gy2)
    return from_remainder_diagram(rho, nu3, gy3, s)

Weight

Created

Jan 29, 2024 at 14:51 by Martin Rubey

Updated

Oct 15, 2024 at 03:00 by Nadia Lafreniere