- FindStat

Identifier

St001561: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [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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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

Created

Jul 11, 2020 at 10:12 by Per Alexandersson

Updated

Jul 11, 2020 at 10:49 by Per Alexandersson