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Identifier

St000490: Set partitions ⟶ ℤ

Values

{{1,2}} => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,3},{2},{4},{5}} => 7

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

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

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

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

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

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

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

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

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

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

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

{{1,4},{2,5},{3}} => 7

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

{{1,4},{2},{3},{5}} => 8

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

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

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

{{1},{2,4},{3},{5}} => 7

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

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

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

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

{{1,5},{2},{3},{4}} => 9

{{1},{2,5},{3},{4}} => 8

{{1},{2},{3,5},{4}} => 7

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

{{1},{2},{3},{4},{5}} => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,4},{3},{5},{6}} => 7

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

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

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Description

The intertwining number of a set partition.
This is defined in [1] as follows: for $\operatorname{int}(a,b) = \{ \min(a,b) < j < \max(a,b) \}$, the block intertwiners of two disjoint sets $A,B$ of integers is given by
$$\{ (a,b) \in A\times B : \operatorname{int}(a,b) \cap A \cup B = \emptyset \}.$$
The intertwining number of a set partition $S$ is now the number of intertwiners of all pairs of blocks of $S$.

References

[1] Ehrenborg, R., Readdy, M. Juggling and applications to $q$-analogues MathSciNet:1417290

Code

def open_interval(a,b):
    return range(min(a,b)+1,max(a,b))

def block_intertwiners(A,B):
    return [ (a,b) for a in A for b in B if all( i not in A and i not in B for i in open_interval(a,b) ) ]

def statistic(S):
    return sum( len( block_intertwiners(A,B) ) for A,B in Subsets(S,2) )

Created

May 16, 2016 at 20:50 by Christian Stump

Updated

Jun 24, 2020 at 10:45 by Tilman Möller