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Identifier
Values
[2] => 1
[1,1] => 0
[3] => 2
[2,1] => 0
[1,1,1] => 0
[4] => 2
[3,1] => 1
[2,2] => -1
[2,1,1] => 0
[1,1,1,1] => 0
[5] => 3
[4,1] => 1
[3,2] => 0
[3,1,1] => 0
[2,2,1] => 0
[2,1,1,1] => 0
[1,1,1,1,1] => 0
[6] => 3
[5,1] => 2
[4,2] => 0
[4,1,1] => 0
[3,3] => 0
[3,2,1] => 0
[3,1,1,1] => 0
[2,2,2] => 1
[2,2,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 0
[7] => 4
[6,1] => 2
[5,2] => 1
[5,1,1] => 0
[4,3] => 0
[4,2,1] => 0
[4,1,1,1] => 0
[3,3,1] => 0
[3,2,2] => 0
[3,2,1,1] => 0
[3,1,1,1,1] => 0
[2,2,2,1] => 0
[2,2,1,1,1] => 0
[2,1,1,1,1,1] => 0
[1,1,1,1,1,1,1] => 0
[8] => 4
[7,1] => 3
[6,2] => 1
[6,1,1] => 0
[5,3] => 2
[5,2,1] => 0
[5,1,1,1] => 0
[4,4] => -2
[4,3,1] => 0
[4,2,2] => 0
[4,2,1,1] => 0
[4,1,1,1,1] => 0
[3,3,2] => 0
[3,3,1,1] => 0
[3,2,2,1] => 0
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 0
[2,2,2,2] => -1
[2,2,2,1,1] => 0
[2,2,1,1,1,1] => 0
[2,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => 0
[9] => 5
[8,1] => 3
[7,2] => 2
[7,1,1] => 0
[6,3] => 2
[6,2,1] => 0
[6,1,1,1] => 0
[5,4] => 0
[5,3,1] => 1
[5,2,2] => -1
[5,2,1,1] => 0
[5,1,1,1,1] => 0
[4,4,1] => -1
[4,3,2] => 0
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 0
[4,1,1,1,1,1] => 0
[3,3,3] => 0
[3,3,2,1] => 0
[3,3,1,1,1] => 0
[3,2,2,2] => 0
[3,2,2,1,1] => 0
[3,2,1,1,1,1] => 0
[3,1,1,1,1,1,1] => 0
[2,2,2,2,1] => 0
[2,2,2,1,1,1] => 0
[2,2,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => 0
[10] => 5
[9,1] => 4
[8,2] => 2
[8,1,1] => 0
[7,3] => 4
[7,2,1] => 0
>>> Load all 270 entries. <<<
[7,1,1,1] => 0
[6,4] => 0
[6,3,1] => 1
[6,2,2] => -1
[6,2,1,1] => 0
[6,1,1,1,1] => 0
[5,5] => 0
[5,4,1] => 0
[5,3,2] => 0
[5,3,1,1] => 0
[5,2,2,1] => 0
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => 0
[4,4,2] => 0
[4,4,1,1] => 0
[4,3,3] => 0
[4,3,2,1] => 0
[4,3,1,1,1] => 0
[4,2,2,2] => 0
[4,2,2,1,1] => 0
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => 0
[3,3,3,1] => 0
[3,3,2,2] => 0
[3,3,2,1,1] => 0
[3,3,1,1,1,1] => 0
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => 0
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 0
[2,2,2,2,2] => 1
[2,2,2,2,1,1] => 0
[2,2,2,1,1,1,1] => 0
[2,2,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 6
[10,1] => 4
[9,2] => 3
[9,1,1] => 0
[8,3] => 4
[8,2,1] => 0
[8,1,1,1] => 0
[7,4] => 2
[7,3,1] => 2
[7,2,2] => -2
[7,2,1,1] => 0
[7,1,1,1,1] => 0
[6,5] => 0
[6,4,1] => 0
[6,3,2] => 0
[6,3,1,1] => 0
[6,2,2,1] => 0
[6,2,1,1,1] => 0
[6,1,1,1,1,1] => 0
[5,5,1] => 0
[5,4,2] => 0
[5,4,1,1] => 0
[5,3,3] => 0
[5,3,2,1] => 0
[5,3,1,1,1] => 0
[5,2,2,2] => 1
[5,2,2,1,1] => 0
[5,2,1,1,1,1] => 0
[5,1,1,1,1,1,1] => 0
[4,4,3] => 0
[4,4,2,1] => 0
[4,4,1,1,1] => 0
[4,3,3,1] => 0
[4,3,2,2] => 0
[4,3,2,1,1] => 0
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 0
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 0
[3,3,3,1,1] => 0
[3,3,2,2,1] => 0
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => 0
[3,2,2,2,2] => 0
[3,2,2,2,1,1] => 0
[3,2,2,1,1,1,1] => 0
[3,2,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,1] => 0
[2,2,2,2,1,1,1] => 0
[2,2,2,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 6
[11,1] => 5
[10,2] => 3
[10,1,1] => 0
[9,3] => 6
[9,2,1] => 0
[9,1,1,1] => 0
[8,4] => 2
[8,3,1] => 2
[8,2,2] => -2
[8,2,1,1] => 0
[8,1,1,1,1] => 0
[7,5] => 3
[7,4,1] => 1
[7,3,2] => 0
[7,3,1,1] => 0
[7,2,2,1] => 0
[7,2,1,1,1] => 0
[7,1,1,1,1,1] => 0
[6,6] => -3
[6,5,1] => 0
[6,4,2] => 0
[6,4,1,1] => 0
[6,3,3] => 0
[6,3,2,1] => 0
[6,3,1,1,1] => 0
[6,2,2,2] => 1
[6,2,2,1,1] => 0
[6,2,1,1,1,1] => 0
[6,1,1,1,1,1,1] => 0
[5,5,2] => 0
[5,5,1,1] => 0
[5,4,3] => 0
[5,4,2,1] => 0
[5,4,1,1,1] => 0
[5,3,3,1] => 0
[5,3,2,2] => 0
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => 0
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 0
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 0
[4,4,4] => 2
[4,4,3,1] => 0
[4,4,2,2] => 0
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => 0
[4,3,3,2] => 0
[4,3,3,1,1] => 0
[4,3,2,2,1] => 0
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => 0
[4,2,2,2,1,1] => 0
[4,2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => 0
[3,3,3,2,1] => 0
[3,3,3,1,1,1] => 0
[3,3,2,2,2] => 0
[3,3,2,2,1,1] => 0
[3,3,2,1,1,1,1] => 0
[3,3,1,1,1,1,1,1] => 0
[3,2,2,2,2,1] => 0
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2] => -1
[2,2,2,2,2,1,1] => 0
[2,2,2,2,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
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Description
Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
$$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$
equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
References
[1] Berkovich, A., Kemal Uncu, A. Variation on a theme of Nathan Fine. New weighted partition identities arXiv:1605.00291
Code
def statistic(pi):
    """
    sage: statistic(Partition([18,12,7,5]))
    12

    Theorem (3.12) of http://arxiv.org/pdf/1605.00291.pdf:
    sage: r=10; DO = [1 for pi in Partitions(r) if len(set(p for p in pi if is_odd(p))) == len([p for p in pi if is_odd(p)])]
    sage: GG1 = [pi for pi in Partitions(r, max_slope=-2) if all(pi[j]-pi[j+1] != 2 for j in range(len(pi)-1) if is_even(pi[j]))]
    sage: sum(statistic(pi) for pi in GG1) == len(DO)
    True
    """
    def delta_even(p):
        if is_even(p):
            return 1
        else:
            return 0

    return (pi[-1] + 1 - delta_even(pi[-1]))/2 * prod((pi[i] - pi[i+1] - delta_even(pi[i]) - delta_even(pi[i+1]))/2 for i in range(len(pi)-1))
Created
May 03, 2016 at 12:34 by Martin Rubey
Updated
May 03, 2016 at 15:48 by Martin Rubey