Identifier
Values
[1] => [1,0] => [1] => 0
[1,1] => [1,0,1,0] => [1,2] => 0
[2] => [1,1,0,0] => [2,1] => 1
[1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 0
[1,2] => [1,0,1,1,0,0] => [1,3,2] => 0
[2,1] => [1,1,0,0,1,0] => [2,1,3] => 0
[3] => [1,1,1,0,0,0] => [3,1,2] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 1
[3,1] => [1,1,1,0,0,0,1,0] => [3,1,2,4] => 0
[4] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,4,5] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,3,4,6] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,3,4,5] => 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,2,4,5,6] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2,4,6,5] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,1,2,5,4,6] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,1,2,6,4,5] => 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,2,3,5,6] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,1,2,3,6,5] => 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => 5
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,5,6] => 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,3,6,4,5,7] => 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,4,5,6] => 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => 0
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,5,6] => 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,3,4,6,7] => 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,3,4,7,6] => 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,3,4,5,7] => 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,3,4,5,6] => 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,2,4,5,7,6] => 0
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => 0
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,5,6] => 0
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6] => 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,4,5,7] => 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,4,5,6] => 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,2,3,5,6,7] => 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,2,3,5,7,6] => 0
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,2,3,6,5,7] => 0
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,2,3,7,5,6] => 0
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Description
Minimum over maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the minimum of this value over all cycles in the permutation.
For example, all permutations with a fixed-point has statistic value 0,
and all permutations of $[n]$ with only one cycle, has statistic value $n-1$.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
bounce path
Description
The bounce path determined by an integer composition.