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Identifier

St001094: Set partitions ⟶ ℤ

Values

{{1}} => 0

{{1,2}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => 15

{{1,2,3,4,5},{6}} => 14

{{1,2,3,4,6},{5}} => 13

{{1,2,3,4},{5,6}} => 14

{{1,2,3,4},{5},{6}} => 12

{{1,2,3,5,6},{4}} => 13

{{1,2,3,5},{4,6}} => 12

{{1,2,3,5},{4},{6}} => 11

{{1,2,3,6},{4,5}} => 13

{{1,2,3},{4,5,6}} => 14

{{1,2,3},{4,5},{6}} => 12

{{1,2,3,6},{4},{5}} => 10

{{1,2,3},{4,6},{5}} => 11

{{1,2,3},{4},{5,6}} => 12

{{1,2,3},{4},{5},{6}} => 9

{{1,2,4,5,6},{3}} => 13

{{1,2,4,5},{3,6}} => 12

{{1,2,4,5},{3},{6}} => 11

{{1,2,4,6},{3,5}} => 11

{{1,2,4},{3,5,6}} => 12

{{1,2,4},{3,5},{6}} => 10

{{1,2,4,6},{3},{5}} => 10

{{1,2,4},{3,6},{5}} => 9

{{1,2,4},{3},{5,6}} => 11

{{1,2,4},{3},{5},{6}} => 8

{{1,2,5,6},{3,4}} => 13

>>> Load all 1487 entries. <<<

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Description

The depth index of a set partition.
For a set partition $\Pi$ of $\{1,\dots,n\}$ with arcs $\mathcal A$, this is $$\sum_{i=1}^{|\mathcal A|} (n-i) - \sum_{j=1}^n depth(j) + \sum_{\alpha\in\mathcal A} depth(\alpha),$$
where the depth of an element $i$ is the number of arcs $(k,\ell)$ with $k < i < \ell$, and the depth of an arc $(i,j)$ is the number of arcs $(k,\ell)$ with $k < i$ and $j < \ell$.

References

[1] Bilen Can, M., Cherniavsky, Y. Stirling Posets arXiv:1801.08231

Code

def depth_vertex(P, i):
    """
    sage: P = SetPartition([{1, 2, 7, 8}, {3, 4, 6}, {5}, {9, 10}])
    sage: [depth_vertex(P, i) for i in range(1, P.size()+1)]
    [0, 0, 1, 1, 2, 1, 0, 0, 0, 0]
    """
    return sum(1 for a in P.arcs() if a[0] < i < a[1])


def depth_arc(P, a):
    """
    sage: P = SetPartition([{1, 2, 7, 8}, {3, 4, 6}, {5}, {9, 10}])
    sage: [(a, depth_arc(P, a)) for a in P.arcs()]
    [((1, 2), 0), ((2, 7), 0), ((7, 8), 0), ((3, 4), 1), ((4, 6), 1), ((9, 10), 0)]
    """
    return sum(1 for b in P.arcs() if b[0] < a[0] and a[1] < b[1])

def statistic(P):
    """
    sage: P = SetPartition([[1,8], [2,5,6,9], [3,7], [4]])
    sage: statistic(P)
    21
    """
    n = P.size()
    A = P.arcs()
    return (sum(n-i for i in range(1, len(A)+1)) -
            sum(depth_vertex(P, i) for i in range(1, n+1)) +
            sum(depth_arc(P, a) for a in A))

Created

Jan 26, 2018 at 08:45 by Martin Rubey

Updated

Feb 01, 2018 at 23:32 by Martin Rubey