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Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations

Images

[1] => [1]

[1,2] => [1,2]

[2,1] => [2,1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 13761 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let

$\pi$ be a permutation. Following Viennot [1], we associate to

$\pi$ a heap of pieces, by considering each decreasing run

$(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of

$\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.

References

[1] Viennot, Gérard X. Heaps of pieces. I. Basic definitions and combinatorial lemmas MathSciNet:0927773
[2] Reading, N. Noncrossing arc diagrams and canonical join representations arXiv:1405.6904

Properties

bijective, graded, involutive

Sage code

def mapping(pi):
    n = len(pi)
    buffer = []
    heap = []
    for i in range(n):
        if not i or pi[i-1] > pi[i]:
            buffer.append(pi[i])
        else:
            heap.append(buffer[::-1])
            buffer = [pi[i]]
    if buffer:
        heap.append(buffer[::-1])
    heap = heap[::-1]
    # bubble sort pieces
    n = len(heap)
    while True:
        try:
            i = next(i for i in range(n-1) if heap[i][0] > heap[i+1][-1])
        except StopIteration:
            break
        heap[i], heap[i+1] = heap[i+1], heap[i]
    sigma = []
    while heap:
        m = []
        i = 0
        for e in heap:
            if (all(min(e) >= max(f) for f in m) or
                all(max(e) <= min(f) for f in m)):
                m.append(e)
                i += 1
            else:
                break
        heap = heap[i:]
        for e in sorted(m, key=min):
            sigma.extend(e[::-1])

    return Permutation(sigma)

Weight

Created

Dec 03, 2021 at 00:11 by Martin Rubey

Updated

Apr 18, 2022 at 18:58 by Nadia Lafreniere