Identifier
Values
[1,2] => [1,0,1,0] => [2,1] => 1
[2,1] => [1,1,0,0] => [1,2] => 0
[1,2,3] => [1,0,1,0,1,0] => [3,2,1] => 1
[1,3,2] => [1,0,1,1,0,0] => [2,3,1] => 1
[2,1,3] => [1,1,0,0,1,0] => [3,1,2] => 1
[2,3,1] => [1,1,0,1,0,0] => [2,1,3] => 1
[3,1,2] => [1,1,1,0,0,0] => [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 2
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 2
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 2
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 2
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 2
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 2
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 2
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 2
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 2
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
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Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.