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Statistic identifier: St000377
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Collection: Integer partitions
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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 $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
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References: [1] Lee, K., Li, L., Loehr, N. A. A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers [[arXiv:1602.01126]]
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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] )
-----------------------------------------------------------------------------
Statistic 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
[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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Created: Feb 06, 2016 at 17:12 by Christian Stump
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Last Updated: Feb 25, 2021 at 20:14 by Martin Rubey