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Identifier

St000211: Set partitions ⟶ ℤ

Values

{{1}} => 0

{{1,2}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.

References

[1] Triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1<=k<=n. OEIS:A008278
[2] De Stavola, D. A Plancherel measure associated to set partitions and its limit arXiv:1612.03061

Code

def statistic(x):
    return x.size()-x.cardinality()

Created

Jun 05, 2014 at 18:58 by Jessica Striker

Updated

Jul 12, 2017 at 09:56 by Christian Stump