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Identifier

Mp00240: Permutations —weak exceedance partition⟶ Set partitions

Images

[1] => {{1}}

[1,2] => {{1},{2}}

[2,1] => {{1,2}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.

Properties

surjective, graded

Sage code

def mapping(pi):
    n = len(pi)
    arcs = [(i, pi(i)) for i in range(1, n+1) if i <= pi(i)]
    return SetPartitions().from_arcs(arcs, n)

Weight

Created

Sep 24, 2021 at 23:19 by Martin Rubey

Updated

Sep 24, 2021 at 23:19 by Martin Rubey