Identifier
Values
=>
Cc0002;cc-rep
[]=>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
[7,3]=>6
[7,2,1]=>5
[7,1,1,1]=>7
[6,4]=>5
[6,3,1]=>4
[6,2,2]=>5
[6,2,1,1]=>6
[6,1,1,1,1]=>6
[5,5]=>6
[5,4,1]=>4
[5,3,2]=>1
[5,3,1,1]=>2
[5,2,2,1]=>3
[5,2,1,1,1]=>5
[5,1,1,1,1,1]=>7
[4,4,2]=>2
[4,4,1,1]=>3
[4,3,3]=>3
[4,3,2,1]=>0
[4,3,1,1,1]=>4
[4,2,2,2]=>4
[4,2,2,1,1]=>5
[4,2,1,1,1,1]=>5
[4,1,1,1,1,1,1]=>7
[3,3,3,1]=>4
[3,3,2,2]=>5
[3,3,2,1,1]=>6
[3,3,1,1,1,1]=>6
[3,2,2,2,1]=>6
[3,2,2,1,1,1]=>7
[3,2,1,1,1,1,1]=>6
[3,1,1,1,1,1,1,1]=>7
[2,2,2,2,2]=>7
[2,2,2,2,1,1]=>8
[2,2,2,1,1,1,1]=>8
[2,2,1,1,1,1,1,1]=>8
[2,1,1,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1,1,1]=>9
[11]=>9
[10,1]=>8
[9,2]=>7
[9,1,1]=>8
[8,3]=>7
[8,2,1]=>6
[8,1,1,1]=>8
[7,4]=>7
[7,3,1]=>5
[7,2,2]=>6
[7,2,1,1]=>7
[7,1,1,1,1]=>8
[6,5]=>6
[6,4,1]=>5
[6,3,2]=>4
[6,3,1,1]=>5
[6,2,2,1]=>6
[6,2,1,1,1]=>6
[6,1,1,1,1,1]=>7
[5,5,1]=>6
[5,4,2]=>2
[5,4,1,1]=>3
[5,3,3]=>3
[5,3,2,1]=>0
[5,3,1,1,1]=>4
[5,2,2,2]=>4
[5,2,2,1,1]=>5
[5,2,1,1,1,1]=>7
[5,1,1,1,1,1,1]=>8
[4,4,3]=>4
[4,4,2,1]=>1
[4,4,1,1,1]=>5
[4,3,3,1]=>2
[4,3,2,2]=>3
[4,3,2,1,1]=>4
[4,3,1,1,1,1]=>5
[4,2,2,2,1]=>6
[4,2,2,1,1,1]=>6
[4,2,1,1,1,1,1]=>6
[4,1,1,1,1,1,1,1]=>8
[3,3,3,2]=>5
[3,3,3,1,1]=>6
[3,3,2,2,1]=>7
[3,3,2,1,1,1]=>7
[3,3,1,1,1,1,1]=>7
[3,2,2,2,2]=>7
[3,2,2,2,1,1]=>8
[3,2,2,1,1,1,1]=>8
[3,2,1,1,1,1,1,1]=>7
[3,1,1,1,1,1,1,1,1]=>8
[2,2,2,2,2,1]=>8
[2,2,2,2,1,1,1]=>9
[2,2,2,1,1,1,1,1]=>9
[2,2,1,1,1,1,1,1,1]=>9
[2,1,1,1,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1,1,1,1]=>10
[12]=>10
[11,1]=>9
[10,2]=>8
[10,1,1]=>9
[9,3]=>8
[9,2,1]=>7
[9,1,1,1]=>9
[8,4]=>8
[8,3,1]=>6
[8,2,2]=>7
[8,2,1,1]=>8
[8,1,1,1,1]=>9
[7,5]=>7
[7,4,1]=>7
[7,3,2]=>5
[7,3,1,1]=>6
[7,2,2,1]=>7
[7,2,1,1,1]=>8
[7,1,1,1,1,1]=>8
[6,6]=>8
[6,5,1]=>6
[6,4,2]=>4
[6,4,1,1]=>5
[6,3,3]=>5
[6,3,2,1]=>4
[6,3,1,1,1]=>6
[6,2,2,2]=>6
[6,2,2,1,1]=>7
[6,2,1,1,1,1]=>7
[6,1,1,1,1,1,1]=>9
[5,5,2]=>5
[5,5,1,1]=>6
[5,4,3]=>3
[5,4,2,1]=>0
[5,4,1,1,1]=>4
[5,3,3,1]=>1
[5,3,2,2]=>2
[5,3,2,1,1]=>3
[5,3,1,1,1,1]=>6
[5,2,2,2,1]=>5
[5,2,2,1,1,1]=>7
[5,2,1,1,1,1,1]=>8
[5,1,1,1,1,1,1,1]=>9
[4,4,4]=>6
[4,4,3,1]=>2
[4,4,2,2]=>3
[4,4,2,1,1]=>4
[4,4,1,1,1,1]=>7
[4,3,3,2]=>4
[4,3,3,1,1]=>5
[4,3,2,2,1]=>6
[4,3,2,1,1,1]=>5
[4,3,1,1,1,1,1]=>6
[4,2,2,2,2]=>7
[4,2,2,2,1,1]=>8
[4,2,2,1,1,1,1]=>7
[4,2,1,1,1,1,1,1]=>7
[4,1,1,1,1,1,1,1,1]=>9
[3,3,3,3]=>7
[3,3,3,2,1]=>6
[3,3,3,1,1,1]=>8
[3,3,2,2,2]=>8
[3,3,2,2,1,1]=>9
[3,3,2,1,1,1,1]=>8
[3,3,1,1,1,1,1,1]=>8
[3,2,2,2,2,1]=>8
[3,2,2,2,1,1,1]=>9
[3,2,2,1,1,1,1,1]=>9
[3,2,1,1,1,1,1,1,1]=>8
[3,1,1,1,1,1,1,1,1,1]=>9
[2,2,2,2,2,2]=>9
[2,2,2,2,2,1,1]=>10
[2,2,2,2,1,1,1,1]=>10
[2,2,2,1,1,1,1,1,1]=>10
[2,2,1,1,1,1,1,1,1,1]=>10
[2,1,1,1,1,1,1,1,1,1,1]=>10
[1,1,1,1,1,1,1,1,1,1,1,1]=>11
[8,5]=>9
[7,5,1]=>7
[7,4,2]=>6
[5,5,3]=>5
[5,4,4]=>6
[5,4,3,1]=>0
[5,4,2,2]=>1
[5,4,2,1,1]=>2
[5,3,3,2]=>2
[5,3,3,1,1]=>3
[5,3,2,2,1]=>4
[4,4,4,1]=>5
[4,4,3,2]=>3
[4,4,3,1,1]=>4
[4,4,2,2,1]=>5
[4,3,3,3]=>6
[4,3,3,2,1]=>6
[3,3,3,3,1]=>7
[3,3,3,2,2]=>8
[9,5]=>10
[8,5,1]=>9
[7,5,2]=>7
[7,4,3]=>7
[5,5,4]=>7
[5,4,3,2]=>0
[5,4,3,1,1]=>1
[5,4,2,2,1]=>2
[5,3,3,2,1]=>3
[5,3,2,2,2]=>6
[4,4,4,2]=>5
[4,4,3,3]=>6
[4,4,3,2,1]=>4
[3,3,3,3,2]=>8
[9,5,1]=>10
[8,5,2]=>9
[7,5,3]=>7
[5,5,5]=>9
[5,4,3,2,1]=>0
[5,3,2,2,2,1]=>8
[4,4,4,3]=>7
[3,3,3,3,3]=>10
[8,5,3]=>9
[7,5,3,1]=>6
[4,4,4,4]=>9
[8,6,3]=>10
[9,6,3]=>12
[8,6,4]=>10
[9,6,4]=>12
[8,5,4,2]=>8
[8,5,5,1]=>11
[7,5,4,3,1]=>2
[8,6,4,2]=>9
[10,6,4]=>13
[10,7,3]=>14
[9,7,4]=>13
[9,5,5,1]=>13
[6,5,4,3,2,1]=>0
[11,7,3]=>15
[9,6,4,3]=>11
[9,6,5,3]=>12
[8,6,5,3,1]=>8
[11,7,5,1]=>16
[9,7,5,3]=>13
[9,7,5,3,1]=>12
[10,7,5,3]=>15
[9,7,5,4,1]=>13
[7,6,5,4,3,2,1]=>0
[10,7,6,4,1]=>16
[9,7,6,4,2]=>12
[10,8,5,4,1]=>17
[10,8,6,4,1]=>17
[9,7,5,5,3,1]=>10
[11,8,6,4,1]=>19
[10,8,6,4,2]=>16
[11,8,6,5,1]=>19
[12,9,7,5,1]=>23
[13,9,7,5,1]=>24
[11,9,7,5,3,1]=>20
[11,8,7,5,4,1]=>19
[8,7,6,5,4,3,2,1]=>0
[11,9,7,5,5,3]=>20
[11,9,7,7,5,3,3]=>18
[11,9,7,6,5,3,1]=>18
[13,11,9,7,5,3,1]=>30
[13,11,9,7,7,5,3,1]=>30
[17,13,11,9,7,5,1]=>46
[15,13,11,9,7,5,3,1]=>42
[29,23,19,17,13,11,7,1]=>103
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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 $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
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
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