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Identifier

St000565: Set partitions ⟶ ℤ

Values

{{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}} => 1

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

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

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

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

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

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

{{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}} => 1

{{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}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 1

{{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}} => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

{{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}} => 1

{{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}} => 1

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

{{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}} => 2

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

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

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

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

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

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

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

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

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

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

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Description

The major index of a set partition.
Let $\pi=B_1/B_2/\dots/B_k$ with $\min B_1<\min B_2<\dots<\min B_k$ a set partition. Let $d_i$ be the number of elements in $B_i$ larger than $\min B_{i+1}$. Then the major index of $\pi$ is $1d_1+2d_2+\dots+(k-1)d_{k-1}$.

References

[1] Sagan, B. E. A maj statistic for set partitions MathSciNet:1087650

Code

def statistic(pi):
    """
    sage: statistic([[1,3,8],[2],[4,6,7],[5,9]])
    8
    """
    pi_sorted = sorted([sorted(b) for b in pi])
    l = len(pi_sorted)-1
    des = [0]*l
    for i in range(l):
        min_next = min(pi_sorted[i+1])
        des[i] = len([e for e in pi_sorted[i] if e > min_next])
        
    return sum(i*d for i, d in enumerate(des, 1))

Created

Aug 06, 2016 at 15:22 by Martin Rubey

Updated

Aug 06, 2016 at 15:22 by Martin Rubey