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Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations

Images

[] => []

[1,0] => [1]

[1,0,1,0] => [1,2]

[1,1,0,0] => [2,1]

[1,0,1,0,1,0] => [1,2,3]

[1,0,1,1,0,0] => [1,3,2]

[1,1,0,0,1,0] => [2,1,3]

[1,1,0,1,0,0] => [2,3,1]

[1,1,1,0,0,0] => [3,2,1]

[1,0,1,0,1,0,1,0] => [1,2,3,4]

[1,0,1,0,1,1,0,0] => [1,2,4,3]

[1,0,1,1,0,0,1,0] => [1,3,2,4]

[1,0,1,1,0,1,0,0] => [1,3,4,2]

[1,0,1,1,1,0,0,0] => [1,4,3,2]

[1,1,0,0,1,0,1,0] => [2,1,3,4]

[1,1,0,0,1,1,0,0] => [2,1,4,3]

[1,1,0,1,0,0,1,0] => [2,3,1,4]

[1,1,0,1,0,1,0,0] => [2,3,4,1]

[1,1,0,1,1,0,0,0] => [2,4,3,1]

[1,1,1,0,0,0,1,0] => [3,2,1,4]

[1,1,1,0,0,1,0,0] => [3,2,4,1]

[1,1,1,0,1,0,0,0] => [4,2,3,1]

[1,1,1,1,0,0,0,0] => [4,3,2,1]

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5]

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4]

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5]

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3]

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3]

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5]

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4]

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5]

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2]

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2]

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5]

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2]

[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2]

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2]

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5]

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4]

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5]

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3]

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3]

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5]

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4]

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5]

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1]

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1]

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5]

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1]

[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1]

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1]

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5]

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4]

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5]

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1]

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1]

[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5]

[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1]

[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1]

[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1]

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5]

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1]

[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1]

[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1]

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1]

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6]

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5]

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6]

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4]

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4]

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6]

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5]

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6]

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3]

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3]

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6]

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3]

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3]

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3]

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6]

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5]

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6]

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4]

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4]

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6]

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5]

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6]

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2]

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2]

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6]

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2]

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2]

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2]

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6]

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5]

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6]

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2]

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2]

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6]

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2]

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2]

>>> Load all 6483 entries. <<<

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

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

References

[1] Stump, C. More bijective Catalan combinatorics on permutations and on signed permutations MathSciNet:3153083 arXiv:0808.2822

Properties

injective, graded

Sage code

def mapping(elt):
    return elt.to_noncrossing_permutation()

Weight

Created

May 02, 2019 at 17:15 by Martin Rubey

Updated

Jan 19, 2020 at 07:28 by Martin Rubey