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Identifier

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions

Images

[1] => [1]

[2] => [1,1]

[1,1] => [2]

[3] => [3]

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

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

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

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

[2,2] => [4]

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

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

[5] => [5]

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

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

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

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

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

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

[6] => [3,3]

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

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

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

[3,3] => [6]

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

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

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

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

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

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

[7] => [7]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8] => [1,1,1,1,1,1,1,1]

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

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

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

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

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

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

[4,4] => [8]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[9] => [9]

[8,1] => [1,1,1,1,1,1,1,1,1]

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

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

[6,3] => [3,3,3]

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,3,3] => [6,3]

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

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

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

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

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

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

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

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

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

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

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

[10] => [5,5]

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

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

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

[7,3] => [7,3]

>>> Load all 1211 entries. <<<

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

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].

References

[1] Pak, I. Partition bijections, a survey MathSciNet:2267263

Properties

bijective, graded

Sage code

def mapping(la):
    mu = []
    nu = []
    for p, m in enumerate(Partition(la).to_exp(), 1):
        if is_even(m):
            mu.extend([2*p]*(m // 2))
        else:
            mu.extend([2*p]*((m-1) // 2))
            nu.append(p)

    i = 0
    while i < len(nu):
        p = nu[i]
        if is_even(p):
            del nu[i]
            nu += [p//2, p//2]
        else:
            i += 1
            
    return Partition(sorted(mu + nu, reverse=True))

Weight

Created

Sep 07, 2023 at 21:27 by Martin Rubey

Updated

Sep 07, 2023 at 21:27 by Martin Rubey