- FindStat

edit this statistic or download as text // json

Identifier

St000378: Integer partitions ⟶ ℤ

Values

[] => 0

[1] => 1

[2] => 2

[1,1] => 1

[3] => 2

[2,1] => 3

[1,1,1] => 1

[4] => 2

[3,1] => 4

[2,2] => 3

[2,1,1] => 2

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

[5] => 2

[4,1] => 3

[3,2] => 5

[3,1,1] => 4

[2,2,1] => 3

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

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

[6] => 2

[5,1] => 3

[4,2] => 5

[4,1,1] => 4

[3,3] => 4

[3,2,1] => 6

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

[2,2,2] => 3

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

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

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

[7] => 2

[6,1] => 3

[5,2] => 4

[5,1,1] => 3

[4,3] => 5

[4,2,1] => 7

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

[3,3,1] => 6

[3,2,2] => 5

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

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

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

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

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

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

[8] => 2

[7,1] => 3

[6,2] => 4

[6,1,1] => 3

[5,3] => 5

[5,2,1] => 5

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

[4,4] => 4

[4,3,1] => 8

[4,2,2] => 7

[4,2,1,1] => 6

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

[3,3,2] => 6

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

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

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

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

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

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

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

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

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

[9] => 2

[8,1] => 3

[7,2] => 4

[7,1,1] => 3

[6,3] => 4

[6,2,1] => 5

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

[5,4] => 5

[5,3,1] => 7

[5,2,2] => 6

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

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

[4,4,1] => 6

[4,3,2] => 9

[4,3,1,1] => 8

[4,2,2,1] => 7

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

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

[3,3,3] => 5

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

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

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

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

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

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

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

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

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

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

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

[10] => 2

[9,1] => 3

[8,2] => 4

[8,1,1] => 3

>>> Load all 745 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 diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].

References

[1] Lee, K., Li, L., Loehr, N. A. A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers arXiv:1602.01126
[2] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics MathSciNet:2371044
[3] Robin Why are the dinv-statistic and the partition length equidistributed? MathOverflow:131484

Code

def statistic(P):
    return sum(1 for c in P.cells() if P.arm_length(*c) - P.leg_length(*c) in [0,1])

Created

Feb 06, 2016 at 17:13 by Christian Stump

Updated

Jan 15, 2024 at 12:24 by Martin Rubey