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Statistic identifier: St001529

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Collection: Integer partitions

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Description: The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. 

In other words, it is the sum of the coefficients in
$$(-1)^{|\lambda|-\ell(\lambda)}\nabla p_\lambda \vert_{q=1,t=1},$$
when expanded in the monomial basis.

Here, $\nabla$ is the linear operator on symmetric functions 
where the modified Macdonald polynomials are eigenvectors. See the Sage documentation for definition and references [[http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sfa.html]]

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References: [1]   Bergeron, F., Garsia, A. M., Haiman, M., Tesler, G. Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions [[DOI:10.4310/maa.1999.v6.n3.a7]]

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Code:


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Statistic values:

[1]         => 1
[2]         => 5
[1,1]       => 4
[3]         => 37
[2,1]       => 55
[1,1,1]     => 23
[4]         => 405
[3,1]       => 587
[2,2]       => 284
[2,1,1]     => 712
[1,1,1,1]   => 206
[5]         => 5251
[4,1]       => 7501
[3,2]       => 7151
[3,1,1]     => 8949
[2,2,1]     => 8719
[2,1,1,1]   => 10103
[1,1,1,1,1] => 2247

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Created: Apr 13, 2020 at 21:26 by Per Alexandersson

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Last Updated: Apr 13, 2020 at 21:26 by Per Alexandersson