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Identifier

St000233: Set partitions ⟶ ℤ

Values

{{1}} => 0

{{1,2}} => 0

{{1},{2}} => 0

{{1,2,3}} => 0

{{1,2},{3}} => 0

{{1,3},{2}} => 0

{{1},{2,3}} => 0

{{1},{2},{3}} => 0

{{1,2,3,4}} => 0

{{1,2,3},{4}} => 0

{{1,2,4},{3}} => 0

{{1,2},{3,4}} => 0

{{1,2},{3},{4}} => 0

{{1,3,4},{2}} => 0

{{1,3},{2,4}} => 0

{{1,3},{2},{4}} => 0

{{1,4},{2,3}} => 1

{{1},{2,3,4}} => 0

{{1},{2,3},{4}} => 0

{{1,4},{2},{3}} => 0

{{1},{2,4},{3}} => 0

{{1},{2},{3,4}} => 0

{{1},{2},{3},{4}} => 0

{{1,2,3,4,5}} => 0

{{1,2,3,4},{5}} => 0

{{1,2,3,5},{4}} => 0

{{1,2,3},{4,5}} => 0

{{1,2,3},{4},{5}} => 0

{{1,2,4,5},{3}} => 0

{{1,2,4},{3,5}} => 0

{{1,2,4},{3},{5}} => 0

{{1,2,5},{3,4}} => 1

{{1,2},{3,4,5}} => 0

{{1,2},{3,4},{5}} => 0

{{1,2,5},{3},{4}} => 0

{{1,2},{3,5},{4}} => 0

{{1,2},{3},{4,5}} => 0

{{1,2},{3},{4},{5}} => 0

{{1,3,4,5},{2}} => 0

{{1,3,4},{2,5}} => 1

{{1,3,4},{2},{5}} => 0

{{1,3,5},{2,4}} => 0

{{1,3},{2,4,5}} => 0

{{1,3},{2,4},{5}} => 0

{{1,3,5},{2},{4}} => 0

{{1,3},{2,5},{4}} => 0

{{1,3},{2},{4,5}} => 0

{{1,3},{2},{4},{5}} => 0

{{1,4,5},{2,3}} => 1

{{1,4},{2,3,5}} => 1

{{1,4},{2,3},{5}} => 1

{{1,5},{2,3,4}} => 2

{{1},{2,3,4,5}} => 0

{{1},{2,3,4},{5}} => 0

{{1,5},{2,3},{4}} => 1

{{1},{2,3,5},{4}} => 0

{{1},{2,3},{4,5}} => 0

{{1},{2,3},{4},{5}} => 0

{{1,4,5},{2},{3}} => 0

{{1,4},{2,5},{3}} => 0

{{1,4},{2},{3,5}} => 0

{{1,4},{2},{3},{5}} => 0

{{1,5},{2,4},{3}} => 1

{{1},{2,4,5},{3}} => 0

{{1},{2,4},{3,5}} => 0

{{1},{2,4},{3},{5}} => 0

{{1,5},{2},{3,4}} => 1

{{1},{2,5},{3,4}} => 1

{{1},{2},{3,4,5}} => 0

{{1},{2},{3,4},{5}} => 0

{{1,5},{2},{3},{4}} => 0

{{1},{2,5},{3},{4}} => 0

{{1},{2},{3,5},{4}} => 0

{{1},{2},{3},{4,5}} => 0

{{1},{2},{3},{4},{5}} => 0

{{1,2,3,4,5,6}} => 0

{{1,2,3,4,5},{6}} => 0

{{1,2,3,4,6},{5}} => 0

{{1,2,3,4},{5,6}} => 0

{{1,2,3,4},{5},{6}} => 0

{{1,2,3,5,6},{4}} => 0

{{1,2,3,5},{4,6}} => 0

{{1,2,3,5},{4},{6}} => 0

{{1,2,3,6},{4,5}} => 1

{{1,2,3},{4,5,6}} => 0

{{1,2,3},{4,5},{6}} => 0

{{1,2,3,6},{4},{5}} => 0

{{1,2,3},{4,6},{5}} => 0

{{1,2,3},{4},{5,6}} => 0

{{1,2,3},{4},{5},{6}} => 0

{{1,2,4,5,6},{3}} => 0

{{1,2,4,5},{3,6}} => 1

{{1,2,4,5},{3},{6}} => 0

{{1,2,4,6},{3,5}} => 0

{{1,2,4},{3,5,6}} => 0

{{1,2,4},{3,5},{6}} => 0

{{1,2,4,6},{3},{5}} => 0

{{1,2,4},{3,6},{5}} => 0

{{1,2,4},{3},{5,6}} => 0

{{1,2,4},{3},{5},{6}} => 0

{{1,2,5,6},{3,4}} => 1

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Description

The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.

References

[1] Chern, B., Diaconis, P., Kane, D. M., Rhoades, R. C. Central Limit Theorems for some Set Partition Statistics arXiv:1502.00938
[2] Kasraoui, A., Zeng, J. Distribution of crossings, nestings and alignments of two edges in matchings and partitions MathSciNet:2212506
[3] Rubey, M., Stump, C. Crossings and nestings in set partitions of classical types MathSciNet:2729369

Code

def statistic(pi):
    pi_sorted = sorted([sorted(b) for b in pi])
    edges = [(a,b) for block in pi_sorted for (a,b) in zip(block, block[1:])]
    count = 0
    for (i1,j1),(i2,j2) in Subsets(edges, 2):
        if i1 < i2 < j2 < j1 or i2 < i1 < j1 < j2:
            count += 1
    return count

Created

Feb 04, 2015 at 11:35 by Christian Stump

Updated

Dec 26, 2017 at 17:19 by Martin Rubey