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Identifier
Values
=>
Cc0002;cc-rep
[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 [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