Identifier
-
Mp00321:
Integer partitions
—2-conjugate⟶
Integer partitions
St000995: Integer partitions ⟶ ℤ
Values
[1] => [1] => 0
[2] => [2] => 2
[1,1] => [1,1] => 0
[3] => [2,1] => 2
[2,1] => [3] => 0
[1,1,1] => [1,1,1] => 0
[4] => [2,2] => 2
[3,1] => [2,1,1] => 2
[2,2] => [4] => 4
[2,1,1] => [3,1] => 0
[1,1,1,1] => [1,1,1,1] => 0
[5] => [2,2,1] => 2
[4,1] => [3,2] => 2
[3,2] => [4,1] => 4
[3,1,1] => [2,1,1,1] => 2
[2,2,1] => [5] => 0
[2,1,1,1] => [3,1,1] => 0
[1,1,1,1,1] => [1,1,1,1,1] => 0
[6] => [2,2,2] => 2
[5,1] => [2,2,1,1] => 2
[4,2] => [4,2] => 4
[4,1,1] => [4,1,1] => 4
[3,3] => [3,2,1] => 2
[3,2,1] => [3,3] => 0
[3,1,1,1] => [2,1,1,1,1] => 2
[2,2,2] => [6] => 6
[2,2,1,1] => [5,1] => 0
[2,1,1,1,1] => [3,1,1,1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[7] => [2,2,2,1] => 2
[6,1] => [3,2,2] => 2
[5,2] => [4,2,1] => 4
[5,1,1] => [2,2,1,1,1] => 2
[4,3] => [4,3] => 4
[4,2,1] => [5,2] => 2
[4,1,1,1] => [4,1,1,1] => 4
[3,3,1] => [3,2,1,1] => 2
[3,2,2] => [6,1] => 6
[3,2,1,1] => [3,3,1] => 0
[3,1,1,1,1] => [2,1,1,1,1,1] => 2
[2,2,2,1] => [7] => 0
[2,2,1,1,1] => [5,1,1] => 0
[2,1,1,1,1,1] => [3,1,1,1,1] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0
[8] => [2,2,2,2] => 2
[7,1] => [2,2,2,1,1] => 2
[6,2] => [4,2,2] => 4
[6,1,1] => [4,2,1,1] => 4
[5,3] => [3,2,2,1] => 2
[5,2,1] => [3,3,2] => 2
[5,1,1,1] => [2,2,1,1,1,1] => 2
[4,4] => [4,4] => 4
[4,3,1] => [4,3,1] => 4
[4,2,2] => [6,2] => 6
[4,2,1,1] => [6,1,1] => 6
[4,1,1,1,1] => [4,1,1,1,1] => 4
[3,3,2] => [5,2,1] => 2
[3,3,1,1] => [3,2,1,1,1] => 2
[3,2,2,1] => [5,3] => 0
[3,2,1,1,1] => [3,3,1,1] => 0
[3,1,1,1,1,1] => [2,1,1,1,1,1,1] => 2
[2,2,2,2] => [8] => 8
[2,2,2,1,1] => [7,1] => 0
[2,2,1,1,1,1] => [5,1,1,1] => 0
[2,1,1,1,1,1,1] => [3,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[9] => [2,2,2,2,1] => 2
[8,1] => [3,2,2,2] => 2
[7,2] => [4,2,2,1] => 4
[7,1,1] => [2,2,2,1,1,1] => 2
[6,3] => [4,3,2] => 4
[6,2,1] => [5,2,2] => 2
[6,1,1,1] => [4,2,1,1,1] => 4
[5,4] => [4,4,1] => 4
[5,3,1] => [3,2,2,1,1] => 2
[5,2,2] => [6,2,1] => 6
[5,2,1,1] => [4,3,1,1] => 4
[5,1,1,1,1] => [2,2,1,1,1,1,1] => 2
[4,4,1] => [5,4] => 4
[4,3,2] => [6,3] => 6
[4,3,1,1] => [5,2,1,1] => 2
[4,2,2,1] => [7,2] => 2
[4,2,1,1,1] => [6,1,1,1] => 6
[4,1,1,1,1,1] => [4,1,1,1,1,1] => 4
[3,3,3] => [3,3,2,1] => 2
[3,3,2,1] => [3,3,3] => 0
[3,3,1,1,1] => [3,2,1,1,1,1] => 2
[3,2,2,2] => [8,1] => 8
[3,2,2,1,1] => [5,3,1] => 0
[3,2,1,1,1,1] => [3,3,1,1,1] => 0
[3,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 2
[2,2,2,2,1] => [9] => 0
[2,2,2,1,1,1] => [7,1,1] => 0
[2,2,1,1,1,1,1] => [5,1,1,1,1] => 0
[2,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 0
[10] => [2,2,2,2,2] => 2
[9,1] => [2,2,2,2,1,1] => 2
[8,2] => [4,2,2,2] => 4
[8,1,1] => [4,2,2,1,1] => 4
[7,3] => [3,2,2,2,1] => 2
>>> Load all 288 entries. <<<
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Description
The largest even part of an integer partition.
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.
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.
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