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Identifier

St001561: Integer partitions ⟶ ℤ

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

[8] => 0

[7,1] => 0

[6,2] => 0

[6,1,1] => 0

[5,3] => 0

[5,2,1] => 0

[5,1,1,1] => 0

[4,4] => 0

[4,3,1] => 0

[4,2,2] => 0

[4,2,1,1] => 96

[4,1,1,1,1] => 3125

[3,3,2] => 3

[3,3,1,1] => 256

[3,2,2,1] => 576

[3,2,1,1,1] => 12500

[3,1,1,1,1,1] => 155520

[2,2,2,2] => 1296

[2,2,2,1,1] => 25000

[2,2,1,1,1,1] => 291600

[2,1,1,1,1,1,1] => 2470629

[1,1,1,1,1,1,1,1] => 16777216

[9] => 0

[8,1] => 0

[7,2] => 0

[7,1,1] => 0

[6,3] => 0

[6,2,1] => 0

[6,1,1,1] => 0

[5,4] => 0

[5,3,1] => 0

[5,2,2] => 0

[5,2,1,1] => 0

[5,1,1,1,1] => 625

[4,4,1] => 0

[4,3,2] => 0

[4,3,1,1] => 64

[4,2,2,1] => 144

[4,2,1,1,1] => 6250

[4,1,1,1,1,1] => 116640

[3,3,3] => 1

[3,3,2,1] => 384

[3,3,1,1,1] => 12500

[3,2,2,2] => 864

[3,2,2,1,1] => 25000

[3,2,1,1,1,1] => 388800

[3,1,1,1,1,1,1] => 4117715

[2,2,2,2,1] => 50000

[2,2,2,1,1,1] => 729000

[2,2,1,1,1,1,1] => 7411887

[2,1,1,1,1,1,1,1] => 58720256

[1,1,1,1,1,1,1,1,1] => 387420489

[10] => 0

[9,1] => 0

[8,2] => 0

[8,1,1] => 0

[7,3] => 0

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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.

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

Code

def statistic(la):
    return prod(falling_factorial(len(la), p)/factorial(p) for p in la)

Created

Jul 11, 2020 at 10:12 by Per Alexandersson

Updated

Jul 27, 2024 at 22:30 by Martin Rubey