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Statistic identifier: St000229
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Collection: Set partitions
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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]
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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]]
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Code:
def statistic(S):
return sum(max(B)-min(B)+1 for B in S)
-----------------------------------------------------------------------------
Statistic values:
{{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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Created: Feb 04, 2015 at 11:18 by Christian Stump
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Last Updated: Sep 29, 2020 at 10:54 by Christian Stump