- FindStat

Identifier

St000377: Integer partitions ⟶ ℤ

Values

[] => 0

[1] => 0

[2] => 0

[1,1] => 1

[3] => 1

[2,1] => 0

[1,1,1] => 2

[4] => 2

[3,1] => 0

[2,2] => 1

[2,1,1] => 2

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

[5] => 3

[4,1] => 2

[3,2] => 0

[3,1,1] => 1

[2,2,1] => 2

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

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

[6] => 4

[5,1] => 3

[4,2] => 1

[4,1,1] => 2

[3,3] => 2

[3,2,1] => 0

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

[2,2,2] => 3

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

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

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

[7] => 5

[6,1] => 4

[5,2] => 3

[5,1,1] => 4

[4,3] => 2

[4,2,1] => 0

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

[3,3,1] => 1

[3,2,2] => 2

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

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

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

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

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

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

[8] => 6

[7,1] => 5

[6,2] => 4

[6,1,1] => 5

[5,3] => 3

[5,2,1] => 3

[5,1,1,1] => 4

[4,4] => 4

[4,3,1] => 0

[4,2,2] => 1

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

[4,1,1,1,1] => 5

[3,3,2] => 2

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

[3,2,2,1] => 4

[3,2,1,1,1] => 4

[3,1,1,1,1,1] => 5

[2,2,2,2] => 5

[2,2,2,1,1] => 6

[2,2,1,1,1,1] => 6

[2,1,1,1,1,1,1] => 6

[1,1,1,1,1,1,1,1] => 7

[9] => 7

[8,1] => 6

[7,2] => 5

[7,1,1] => 6

[6,3] => 5

[6,2,1] => 4

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

[5,4] => 4

[5,3,1] => 2

[5,2,2] => 3

[5,2,1,1] => 4

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

[4,4,1] => 3

[4,3,2] => 0

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

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

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

[4,1,1,1,1,1] => 6

[3,3,3] => 4

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

[3,3,1,1,1] => 5

[3,2,2,2] => 5

[3,2,2,1,1] => 6

[3,2,1,1,1,1] => 5

[3,1,1,1,1,1,1] => 6

[2,2,2,2,1] => 6

[2,2,2,1,1,1] => 7

[2,2,1,1,1,1,1] => 7

[2,1,1,1,1,1,1,1] => 7

[1,1,1,1,1,1,1,1,1] => 8

[10] => 8

[9,1] => 7

[8,2] => 6

[8,1,1] => 7

>>> Load all 360 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

click to show known generating functions

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 $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.

References

[1] Lee, K., Li, L., Loehr, N. A. A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers arXiv:1602.01126

Code

def statistic(P):
    return sum( 1 for c in P.cells() if P.arm_length(*c)-P.leg_length(*c) not in [0,1] )

Created

Feb 06, 2016 at 17:12 by Christian Stump

Updated

Feb 25, 2021 at 20:14 by Martin Rubey