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Identifier

St001843: 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}} => 2

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

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

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

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

{{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}} => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 2

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

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

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

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

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

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

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

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

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

{{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}} => 4

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

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

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

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

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

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

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

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

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

{{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}} => 2

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

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

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

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

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

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

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

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

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

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

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Description

The Z-index of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The Z-index of $w$ equals
$$ \sum_{i < j} w_{i,j}, $$
where $w_{i,j}$ is the word obtained from $w$ by removing all letters different from $i$ and $j$.

References

[1] Liu, S.-H. Mahonian and Euler-Mahonian statistics for set partitions arXiv:2202.02089

Code

def to_restricted_growth_word_blocks_max(self):
    w = [0] * self.size()
    for i, B in enumerate(sorted(self, key=lambda B: max(B))):
        for j in B:
            w[j-1] = i
    return w

def statistic(p):
    w = to_restricted_growth_word_blocks_max(p)
    return sum(Word(e for e in w if e in [i, j]).major_index() for j in range(len(w)) for i in range(j))

Created

Oct 06, 2022 at 15:30 by Martin Rubey

Updated

Oct 06, 2022 at 15:30 by Martin Rubey