Identifier
-
Mp00322:
Integer partitions
—Loehr-Warrington⟶
Integer partitions
St000377: Integer partitions ⟶ ℤ
Values
[1] => [1] => 0
[2] => [1,1] => 1
[1,1] => [2] => 0
[3] => [1,1,1] => 2
[2,1] => [3] => 1
[1,1,1] => [2,1] => 0
[4] => [1,1,1,1] => 3
[3,1] => [2,1,1] => 2
[2,2] => [4] => 2
[2,1,1] => [2,2] => 1
[1,1,1,1] => [3,1] => 0
[5] => [1,1,1,1,1] => 4
[4,1] => [2,1,1,1] => 3
[3,2] => [5] => 3
[3,1,1] => [4,1] => 2
[2,2,1] => [2,2,1] => 2
[2,1,1,1] => [3,1,1] => 1
[1,1,1,1,1] => [3,2] => 0
[6] => [1,1,1,1,1,1] => 5
[5,1] => [2,1,1,1,1] => 4
[4,2] => [2,2,1,1] => 4
[4,1,1] => [3,1,1,1] => 3
[3,3] => [6] => 4
[3,2,1] => [5,1] => 3
[3,1,1,1] => [3,3] => 2
[2,2,2] => [2,2,2] => 3
[2,2,1,1] => [4,1,1] => 2
[2,1,1,1,1] => [4,2] => 1
[1,1,1,1,1,1] => [3,2,1] => 0
[7] => [1,1,1,1,1,1,1] => 6
[6,1] => [2,1,1,1,1,1] => 5
[5,2] => [2,2,1,1,1] => 5
[5,1,1] => [3,1,1,1,1] => 4
[4,3] => [7] => 5
[4,2,1] => [5,1,1] => 4
[4,1,1,1] => [3,2,1,1] => 3
[3,3,1] => [6,1] => 4
[3,2,2] => [2,2,2,1] => 4
[3,2,1,1] => [5,2] => 3
[3,1,1,1,1] => [3,2,2] => 2
[2,2,2,1] => [4,1,1,1] => 3
[2,2,1,1,1] => [4,3] => 2
[2,1,1,1,1,1] => [3,3,1] => 1
[1,1,1,1,1,1,1] => [4,2,1] => 0
[8] => [1,1,1,1,1,1,1,1] => 7
[7,1] => [2,1,1,1,1,1,1] => 6
[6,2] => [2,2,1,1,1,1] => 6
[6,1,1] => [3,1,1,1,1,1] => 5
[5,3] => [2,2,2,1,1] => 6
[5,2,1] => [4,1,1,1,1] => 5
[5,1,1,1] => [3,2,1,1,1] => 4
[4,4] => [8] => 6
[4,3,1] => [7,1] => 5
[4,2,2] => [6,1,1] => 5
[4,2,1,1] => [4,4] => 4
[4,1,1,1,1] => [5,2,1] => 3
[3,3,2] => [2,2,2,2] => 5
[3,3,1,1] => [6,2] => 4
[3,2,2,1] => [3,2,2,1] => 4
[3,2,1,1,1] => [3,3,1,1] => 3
[3,1,1,1,1,1] => [4,2,1,1] => 2
[2,2,2,2] => [5,1,1,1] => 4
[2,2,2,1,1] => [5,3] => 3
[2,2,1,1,1,1] => [3,3,2] => 2
[2,1,1,1,1,1,1] => [4,2,2] => 1
[1,1,1,1,1,1,1,1] => [4,3,1] => 0
[9] => [1,1,1,1,1,1,1,1,1] => 8
[8,1] => [2,1,1,1,1,1,1,1] => 7
[7,2] => [2,2,1,1,1,1,1] => 7
[7,1,1] => [3,1,1,1,1,1,1] => 6
[6,3] => [2,2,2,1,1,1] => 7
[6,2,1] => [4,1,1,1,1,1] => 6
[6,1,1,1] => [3,2,1,1,1,1] => 5
[5,4] => [9] => 7
[5,3,1] => [3,2,2,1,1] => 6
[5,2,2] => [6,1,1,1] => 6
[5,2,1,1] => [3,3,1,1,1] => 5
[5,1,1,1,1] => [4,2,1,1,1] => 4
[4,4,1] => [8,1] => 6
[4,3,2] => [7,1,1] => 6
[4,3,1,1] => [7,2] => 5
[4,2,2,1] => [6,3] => 5
[4,2,1,1,1] => [6,2,1] => 4
[4,1,1,1,1,1] => [4,4,1] => 3
[3,3,3] => [2,2,2,2,1] => 6
[3,3,2,1] => [3,2,2,2] => 5
[3,3,1,1,1] => [5,2,1,1] => 4
[3,2,2,2] => [5,1,1,1,1] => 5
[3,2,2,1,1] => [3,3,3] => 4
[3,2,1,1,1,1] => [5,2,2] => 3
[3,1,1,1,1,1,1] => [5,3,1] => 2
[2,2,2,2,1] => [5,4] => 4
[2,2,2,1,1,1] => [3,3,2,1] => 3
[2,2,1,1,1,1,1] => [4,2,2,1] => 2
[2,1,1,1,1,1,1,1] => [4,3,1,1] => 1
[1,1,1,1,1,1,1,1,1] => [4,3,2] => 0
[10] => [1,1,1,1,1,1,1,1,1,1] => 9
[9,1] => [2,1,1,1,1,1,1,1,1] => 8
[8,2] => [2,2,1,1,1,1,1,1] => 8
[8,1,1] => [3,1,1,1,1,1,1,1] => 7
[7,3] => [2,2,2,1,1,1,1] => 8
>>> Load all 318 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Map
Loehr-Warrington
Description
Return a partition whose diagonal inversion number is the length of the preimage.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!