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Identifier

St001641: Set partitions ⟶ ℤ

Values

{{1}} => 0

{{1,2}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of ascent tops in the flattened set partition such that all smaller elements appear before.
Let $P$ be a set partition. The flattened set partition is the permutation obtained by sorting the set of blocks of $P$ according to their minimal element and the elements in each block in increasing order.
Given a set partition $P$, this statistic is the binary logarithm of the number of set partitions that flatten to the same permutation as $P$.

References

[1] Nabawanda, O., Rakotondrajao, F. The sets of flattened partitions with forbidden patterns arXiv:2011.07304

Code

def flatten(P):
    return tuple(e for B in sorted(P, key=min) for e in sorted(list(B)))

def statistic(P):
    p = flatten(P)
    return sum(1 for i in range(len(p)-1) if p[i] < p[i+1] and set(range(1, p[i+1])).issubset(sorted(p[:i+1])))

Created

Nov 17, 2020 at 11:00 by Martin Rubey

Updated

Nov 17, 2020 at 11:00 by Martin Rubey