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Identifier

St000496: Set partitions ⟶ ℤ

Values

{{1}} => 0

{{1,2}} => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 1

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

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

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

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

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

{{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}} => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a rcs (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.

References

[1] Steingrimsson, E. Statistics on ordered partitions of sets arXiv:math/0605670

Code

def statistic(S):
    S = sorted( sorted(B) for B in S )
    n = sum( len(B) for B in S )
    rcs = 0
    for k in range(len(S)):
        j = S[k][-1]
        for l in range(k):
            for i in S[l]:
                if i > j:
                    rcs += 1
    return rcs

Created

May 18, 2016 at 12:26 by Christian Stump

Updated

Dec 28, 2017 at 20:39 by Martin Rubey