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Statistic identifier: St001561

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Collection: Integer partitions

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Description: The value of the elementary symmetric function evaluated at 1.

The statistic is $e_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k=1$,
where $\lambda$ has $k$ parts.

Thus, the statistic is equal to $\prod_{j=1}^k \frac{(k)_{\lambda_j}}{\lambda_j!}$
where $\lambda$ has $k$ parts.

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References: [1]   Stanley, R. P. Enumerative combinatorics. Vol. 2 [[MathSciNet:1676282]]
[2]   Rosas, M. H. Specializations of MacMahon symmetric functions and the polynomial algebra [[DOI:10.1016/s0012-365x(01)00263-1]]

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Code:


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Statistic values:

[1]             => 1
[2]             => 0
[1,1]           => 4
[3]             => 0
[2,1]           => 2
[1,1,1]         => 27
[4]             => 0
[3,1]           => 0
[2,2]           => 1
[2,1,1]         => 27
[1,1,1,1]       => 256
[5]             => 0
[4,1]           => 0
[3,2]           => 0
[3,1,1]         => 9
[2,2,1]         => 27
[2,1,1,1]       => 384
[1,1,1,1,1]     => 3125
[6]             => 0
[5,1]           => 0
[4,2]           => 0
[4,1,1]         => 0
[3,3]           => 0
[3,2,1]         => 9
[3,1,1,1]       => 256
[2,2,2]         => 27
[2,2,1,1]       => 576
[2,1,1,1,1]     => 6250
[1,1,1,1,1,1]   => 46656
[7]             => 0
[6,1]           => 0
[5,2]           => 0
[5,1,1]         => 0
[4,3]           => 0
[4,2,1]         => 0
[4,1,1,1]       => 64
[3,3,1]         => 3
[3,2,2]         => 9
[3,2,1,1]       => 384
[3,1,1,1,1]     => 6250
[2,2,2,1]       => 864
[2,2,1,1,1]     => 12500
[2,1,1,1,1,1]   => 116640
[1,1,1,1,1,1,1] => 823543

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Created: Jul 11, 2020 at 10:12 by Per Alexandersson

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Last Updated: Jul 11, 2020 at 10:49 by Per Alexandersson