Identifier
Values
[1] => [1] => 1
[2] => [2] => 1
[1,1] => [1,1] => 4
[3] => [2,1] => 6
[2,1] => [3] => 1
[1,1,1] => [1,1,1] => 27
[4] => [2,2] => 9
[3,1] => [2,1,1] => 54
[2,2] => [4] => 1
[2,1,1] => [3,1] => 8
[1,1,1,1] => [1,1,1,1] => 256
[5] => [2,2,1] => 108
[4,1] => [3,2] => 12
[3,2] => [4,1] => 10
[3,1,1] => [2,1,1,1] => 640
[2,2,1] => [5] => 1
[2,1,1,1] => [3,1,1] => 90
[1,1,1,1,1] => [1,1,1,1,1] => 3125
[6] => [2,2,2] => 216
[5,1] => [2,2,1,1] => 1600
[4,2] => [4,2] => 15
[4,1,1] => [4,1,1] => 135
[3,3] => [3,2,1] => 180
[3,2,1] => [3,3] => 16
[3,1,1,1] => [2,1,1,1,1] => 9375
[2,2,2] => [6] => 1
[2,2,1,1] => [5,1] => 12
[2,1,1,1,1] => [3,1,1,1] => 1280
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 46656
[7] => [2,2,2,1] => 4000
[6,1] => [3,2,2] => 360
[5,2] => [4,2,1] => 270
[5,1,1] => [2,2,1,1,1] => 28125
[4,3] => [4,3] => 20
[4,2,1] => [5,2] => 18
[4,1,1,1] => [4,1,1,1] => 2240
[3,3,1] => [3,2,1,1] => 3200
[3,2,2] => [6,1] => 14
[3,2,1,1] => [3,3,1] => 300
[3,1,1,1,1] => [2,1,1,1,1,1] => 163296
[2,2,2,1] => [7] => 1
[2,2,1,1,1] => [5,1,1] => 189
[2,1,1,1,1,1] => [3,1,1,1,1] => 21875
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 823543
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Description
The value of the complete homogeneous symmetric function evaluated at 1.
The statistic is $h_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$,
where $\lambda$ has $k$ parts.
Map
2-conjugate
Description
Return a partition with the same number of odd parts and number of even parts interchanged with the number of cells with zero leg and odd arm length.
This is a special case of an involution that preserves the sequence of non-zero remainders of the parts under division by $s$ and interchanges the number of parts divisible by $s$ and the number of cells with zero leg length and arm length congruent to $s-1$ modulo $s$.
In particular, for $s=1$ the involution is conjugation, hence the name.