- FindStat

edit this map or download as text // json

Identifier

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations

Images

[] => []

[1,0] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 6518 entries. <<<

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

Description

Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.

References

[1] Elizalde, S., Pak, I. Bijections for refined restricted permutations MathSciNet:2046080
[2] Claesson, A., Kitaev, S. Classification of bijections between 321- and 132-avoiding permutations MathSciNet:2465405
[3] Knuth, D. E. The art of computer programming. Vol. 3 MathSciNet:3077154

Properties

injective, graded

Sage code

def mapping(elt):
    A, B = elt.to_pair_of_standard_tableaux()
    return RSK_inverse(A, B, output='permutation')

Weight

Created

Jan 19, 2020 at 07:28 by FindStatCrew

Updated

Jan 19, 2020 at 07:28 by Martin Rubey