Identifier
- St000229: Set partitions ⟶ ℤ
Values
=>
Cc0009;cc-rep
{{1}}=>1
{{1,2}}=>2
{{1},{2}}=>2
{{1,2,3}}=>3
{{1,2},{3}}=>3
{{1,3},{2}}=>4
{{1},{2,3}}=>3
{{1},{2},{3}}=>3
{{1,2,3,4}}=>4
{{1,2,3},{4}}=>4
{{1,2,4},{3}}=>5
{{1,2},{3,4}}=>4
{{1,2},{3},{4}}=>4
{{1,3,4},{2}}=>5
{{1,3},{2,4}}=>6
{{1,3},{2},{4}}=>5
{{1,4},{2,3}}=>6
{{1},{2,3,4}}=>4
{{1},{2,3},{4}}=>4
{{1,4},{2},{3}}=>6
{{1},{2,4},{3}}=>5
{{1},{2},{3,4}}=>4
{{1},{2},{3},{4}}=>4
{{1,2,3,4,5}}=>5
{{1,2,3,4},{5}}=>5
{{1,2,3,5},{4}}=>6
{{1,2,3},{4,5}}=>5
{{1,2,3},{4},{5}}=>5
{{1,2,4,5},{3}}=>6
{{1,2,4},{3,5}}=>7
{{1,2,4},{3},{5}}=>6
{{1,2,5},{3,4}}=>7
{{1,2},{3,4,5}}=>5
{{1,2},{3,4},{5}}=>5
{{1,2,5},{3},{4}}=>7
{{1,2},{3,5},{4}}=>6
{{1,2},{3},{4,5}}=>5
{{1,2},{3},{4},{5}}=>5
{{1,3,4,5},{2}}=>6
{{1,3,4},{2,5}}=>8
{{1,3,4},{2},{5}}=>6
{{1,3,5},{2,4}}=>8
{{1,3},{2,4,5}}=>7
{{1,3},{2,4},{5}}=>7
{{1,3,5},{2},{4}}=>7
{{1,3},{2,5},{4}}=>8
{{1,3},{2},{4,5}}=>6
{{1,3},{2},{4},{5}}=>6
{{1,4,5},{2,3}}=>7
{{1,4},{2,3,5}}=>8
{{1,4},{2,3},{5}}=>7
{{1,5},{2,3,4}}=>8
{{1},{2,3,4,5}}=>5
{{1},{2,3,4},{5}}=>5
{{1,5},{2,3},{4}}=>8
{{1},{2,3,5},{4}}=>6
{{1},{2,3},{4,5}}=>5
{{1},{2,3},{4},{5}}=>5
{{1,4,5},{2},{3}}=>7
{{1,4},{2,5},{3}}=>9
{{1,4},{2},{3,5}}=>8
{{1,4},{2},{3},{5}}=>7
{{1,5},{2,4},{3}}=>9
{{1},{2,4,5},{3}}=>6
{{1},{2,4},{3,5}}=>7
{{1},{2,4},{3},{5}}=>6
{{1,5},{2},{3,4}}=>8
{{1},{2,5},{3,4}}=>7
{{1},{2},{3,4,5}}=>5
{{1},{2},{3,4},{5}}=>5
{{1,5},{2},{3},{4}}=>8
{{1},{2,5},{3},{4}}=>7
{{1},{2},{3,5},{4}}=>6
{{1},{2},{3},{4,5}}=>5
{{1},{2},{3},{4},{5}}=>5
{{1,2,3,4,5,6}}=>6
{{1,2,3,4,5},{6}}=>6
{{1,2,3,4,6},{5}}=>7
{{1,2,3,4},{5,6}}=>6
{{1,2,3,4},{5},{6}}=>6
{{1,2,3,5,6},{4}}=>7
{{1,2,3,5},{4,6}}=>8
{{1,2,3,5},{4},{6}}=>7
{{1,2,3,6},{4,5}}=>8
{{1,2,3},{4,5,6}}=>6
{{1,2,3},{4,5},{6}}=>6
{{1,2,3,6},{4},{5}}=>8
{{1,2,3},{4,6},{5}}=>7
{{1,2,3},{4},{5,6}}=>6
{{1,2,3},{4},{5},{6}}=>6
{{1,2,4,5,6},{3}}=>7
{{1,2,4,5},{3,6}}=>9
{{1,2,4,5},{3},{6}}=>7
{{1,2,4,6},{3,5}}=>9
{{1,2,4},{3,5,6}}=>8
{{1,2,4},{3,5},{6}}=>8
{{1,2,4,6},{3},{5}}=>8
{{1,2,4},{3,6},{5}}=>9
{{1,2,4},{3},{5,6}}=>7
{{1,2,4},{3},{5},{6}}=>7
{{1,2,5,6},{3,4}}=>8
{{1,2,5},{3,4,6}}=>9
{{1,2,5},{3,4},{6}}=>8
{{1,2,6},{3,4,5}}=>9
{{1,2},{3,4,5,6}}=>6
{{1,2},{3,4,5},{6}}=>6
{{1,2,6},{3,4},{5}}=>9
{{1,2},{3,4,6},{5}}=>7
{{1,2},{3,4},{5,6}}=>6
{{1,2},{3,4},{5},{6}}=>6
{{1,2,5,6},{3},{4}}=>8
{{1,2,5},{3,6},{4}}=>10
{{1,2,5},{3},{4,6}}=>9
{{1,2,5},{3},{4},{6}}=>8
{{1,2,6},{3,5},{4}}=>10
{{1,2},{3,5,6},{4}}=>7
{{1,2},{3,5},{4,6}}=>8
{{1,2},{3,5},{4},{6}}=>7
{{1,2,6},{3},{4,5}}=>9
{{1,2},{3,6},{4,5}}=>8
{{1,2},{3},{4,5,6}}=>6
{{1,2},{3},{4,5},{6}}=>6
{{1,2,6},{3},{4},{5}}=>9
{{1,2},{3,6},{4},{5}}=>8
{{1,2},{3},{4,6},{5}}=>7
{{1,2},{3},{4},{5,6}}=>6
{{1,2},{3},{4},{5},{6}}=>6
{{1,3,4,5,6},{2}}=>7
{{1,3,4,5},{2,6}}=>10
{{1,3,4,5},{2},{6}}=>7
{{1,3,4,6},{2,5}}=>10
{{1,3,4},{2,5,6}}=>9
{{1,3,4},{2,5},{6}}=>9
{{1,3,4,6},{2},{5}}=>8
{{1,3,4},{2,6},{5}}=>10
{{1,3,4},{2},{5,6}}=>7
{{1,3,4},{2},{5},{6}}=>7
{{1,3,5,6},{2,4}}=>9
{{1,3,5},{2,4,6}}=>10
{{1,3,5},{2,4},{6}}=>9
{{1,3,6},{2,4,5}}=>10
{{1,3},{2,4,5,6}}=>8
{{1,3},{2,4,5},{6}}=>8
{{1,3,6},{2,4},{5}}=>10
{{1,3},{2,4,6},{5}}=>9
{{1,3},{2,4},{5,6}}=>8
{{1,3},{2,4},{5},{6}}=>8
{{1,3,5,6},{2},{4}}=>8
{{1,3,5},{2,6},{4}}=>11
{{1,3,5},{2},{4,6}}=>9
{{1,3,5},{2},{4},{6}}=>8
{{1,3,6},{2,5},{4}}=>11
{{1,3},{2,5,6},{4}}=>9
{{1,3},{2,5},{4,6}}=>10
{{1,3},{2,5},{4},{6}}=>9
{{1,3,6},{2},{4,5}}=>9
{{1,3},{2,6},{4,5}}=>10
{{1,3},{2},{4,5,6}}=>7
{{1,3},{2},{4,5},{6}}=>7
{{1,3,6},{2},{4},{5}}=>9
{{1,3},{2,6},{4},{5}}=>10
{{1,3},{2},{4,6},{5}}=>8
{{1,3},{2},{4},{5,6}}=>7
{{1,3},{2},{4},{5},{6}}=>7
{{1,4,5,6},{2,3}}=>8
{{1,4,5},{2,3,6}}=>10
{{1,4,5},{2,3},{6}}=>8
{{1,4,6},{2,3,5}}=>10
{{1,4},{2,3,5,6}}=>9
{{1,4},{2,3,5},{6}}=>9
{{1,4,6},{2,3},{5}}=>9
{{1,4},{2,3,6},{5}}=>10
{{1,4},{2,3},{5,6}}=>8
{{1,4},{2,3},{5},{6}}=>8
{{1,5,6},{2,3,4}}=>9
{{1,5},{2,3,4,6}}=>10
{{1,5},{2,3,4},{6}}=>9
{{1,6},{2,3,4,5}}=>10
{{1},{2,3,4,5,6}}=>6
{{1},{2,3,4,5},{6}}=>6
{{1,6},{2,3,4},{5}}=>10
{{1},{2,3,4,6},{5}}=>7
{{1},{2,3,4},{5,6}}=>6
{{1},{2,3,4},{5},{6}}=>6
{{1,5,6},{2,3},{4}}=>9
{{1,5},{2,3,6},{4}}=>11
{{1,5},{2,3},{4,6}}=>10
{{1,5},{2,3},{4},{6}}=>9
{{1,6},{2,3,5},{4}}=>11
{{1},{2,3,5,6},{4}}=>7
{{1},{2,3,5},{4,6}}=>8
{{1},{2,3,5},{4},{6}}=>7
{{1,6},{2,3},{4,5}}=>10
{{1},{2,3,6},{4,5}}=>8
{{1},{2,3},{4,5,6}}=>6
{{1},{2,3},{4,5},{6}}=>6
{{1,6},{2,3},{4},{5}}=>10
{{1},{2,3,6},{4},{5}}=>8
{{1},{2,3},{4,6},{5}}=>7
{{1},{2,3},{4},{5,6}}=>6
{{1},{2,3},{4},{5},{6}}=>6
{{1,4,5,6},{2},{3}}=>8
{{1,4,5},{2,6},{3}}=>11
{{1,4,5},{2},{3,6}}=>10
{{1,4,5},{2},{3},{6}}=>8
{{1,4,6},{2,5},{3}}=>11
{{1,4},{2,5,6},{3}}=>10
{{1,4},{2,5},{3,6}}=>12
{{1,4},{2,5},{3},{6}}=>10
{{1,4,6},{2},{3,5}}=>10
{{1,4},{2,6},{3,5}}=>12
{{1,4},{2},{3,5,6}}=>9
{{1,4},{2},{3,5},{6}}=>9
{{1,4,6},{2},{3},{5}}=>9
{{1,4},{2,6},{3},{5}}=>11
{{1,4},{2},{3,6},{5}}=>10
{{1,4},{2},{3},{5,6}}=>8
{{1,4},{2},{3},{5},{6}}=>8
{{1,5,6},{2,4},{3}}=>10
{{1,5},{2,4,6},{3}}=>11
{{1,5},{2,4},{3,6}}=>12
{{1,5},{2,4},{3},{6}}=>10
{{1,6},{2,4,5},{3}}=>11
{{1},{2,4,5,6},{3}}=>7
{{1},{2,4,5},{3,6}}=>9
{{1},{2,4,5},{3},{6}}=>7
{{1,6},{2,4},{3,5}}=>12
{{1},{2,4,6},{3,5}}=>9
{{1},{2,4},{3,5,6}}=>8
{{1},{2,4},{3,5},{6}}=>8
{{1,6},{2,4},{3},{5}}=>11
{{1},{2,4,6},{3},{5}}=>8
{{1},{2,4},{3,6},{5}}=>9
{{1},{2,4},{3},{5,6}}=>7
{{1},{2,4},{3},{5},{6}}=>7
{{1,5,6},{2},{3,4}}=>9
{{1,5},{2,6},{3,4}}=>12
{{1,5},{2},{3,4,6}}=>10
{{1,5},{2},{3,4},{6}}=>9
{{1,6},{2,5},{3,4}}=>12
{{1},{2,5,6},{3,4}}=>8
{{1},{2,5},{3,4,6}}=>9
{{1},{2,5},{3,4},{6}}=>8
{{1,6},{2},{3,4,5}}=>10
{{1},{2,6},{3,4,5}}=>9
{{1},{2},{3,4,5,6}}=>6
{{1},{2},{3,4,5},{6}}=>6
{{1,6},{2},{3,4},{5}}=>10
{{1},{2,6},{3,4},{5}}=>9
{{1},{2},{3,4,6},{5}}=>7
{{1},{2},{3,4},{5,6}}=>6
{{1},{2},{3,4},{5},{6}}=>6
{{1,5,6},{2},{3},{4}}=>9
{{1,5},{2,6},{3},{4}}=>12
{{1,5},{2},{3,6},{4}}=>11
{{1,5},{2},{3},{4,6}}=>10
{{1,5},{2},{3},{4},{6}}=>9
{{1,6},{2,5},{3},{4}}=>12
{{1},{2,5,6},{3},{4}}=>8
{{1},{2,5},{3,6},{4}}=>10
{{1},{2,5},{3},{4,6}}=>9
{{1},{2,5},{3},{4},{6}}=>8
{{1,6},{2},{3,5},{4}}=>11
{{1},{2,6},{3,5},{4}}=>10
{{1},{2},{3,5,6},{4}}=>7
{{1},{2},{3,5},{4,6}}=>8
{{1},{2},{3,5},{4},{6}}=>7
{{1,6},{2},{3},{4,5}}=>10
{{1},{2,6},{3},{4,5}}=>9
{{1},{2},{3,6},{4,5}}=>8
{{1},{2},{3},{4,5,6}}=>6
{{1},{2},{3},{4,5},{6}}=>6
{{1,6},{2},{3},{4},{5}}=>10
{{1},{2,6},{3},{4},{5}}=>9
{{1},{2},{3,6},{4},{5}}=>8
{{1},{2},{3},{4,6},{5}}=>7
{{1},{2},{3},{4},{5,6}}=>6
{{1},{2},{3},{4},{5},{6}}=>6
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Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called dimension index in [2]
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called dimension index in [2]
References
[1] Chern, B., Diaconis, P., Kane, D. M., Rhoades, R. C. Central Limit Theorems for some Set Partition Statistics arXiv:1502.00938
[2] Grubb, T., Rajasekaran, F. Set Partition Patterns and the Dimension Index arXiv:2009.00650
[2] Grubb, T., Rajasekaran, F. Set Partition Patterns and the Dimension Index arXiv:2009.00650
Code
def statistic(S):
return sum(max(B)-min(B)+1 for B in S)
Created
Feb 04, 2015 at 11:18 by Christian Stump
Updated
Sep 29, 2020 at 10:54 by Christian Stump
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