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Identifier

Mp00257: Permutations —Alexandersson Kebede⟶ Permutations

Images

[1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7576 entries. <<<

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

Description

Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation

$\pi$ of length

$n$ . The mapping looks for a smallest odd integer

$i\in[n-1]$ such that swapping the entries

$\pi(i)$ and

$\pi(i+1)$ preserves the set of right-to-left minima. Otherwise,

$\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly

$\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in

$S_n$ fixed under this map, with exactly

$k$ right-to-left minima, see Lemma 35 in [1].

References

[1] Alexandersson, P., Getachew Kebede, F. An involution on derangements preserving excedances and right-to-left minima arXiv:2105.08455

Properties

bijective, graded, involutive

Sage code

def rlmin(pi):
    n = len(pi)
    m = n+1
    for i in range(n):
        if pi[n-1-i] < m:
            yield pi[n-1-i]
            m = pi[n-1-i]

def mapping(pi):
    """
    sage: all(mapping(mapping(pi)) == pi for pi in Permutations(6))
    True

    sage: len([pi for pi in Permutations(7) if mapping(pi) == pi])
    8

    """
    s = list(rlmin(pi))
    for i in range(0, len(pi)-1, 2):
        sigma = pi[:]
        sigma[i] = pi[i+1]
        sigma[i+1] = pi[i]
        if s == list(rlmin(sigma)):
            return Permutation(sigma)
    return pi

Weight

Created

Jul 21, 2022 at 07:33 by Per Alexandersson and Frether Getachew Kebede

Updated

Aug 07, 2023 at 20:19 by Nadia Lafreniere