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 $4$.
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
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