Identifier
Values
[1,0,1,0] => [1,2] => 1
[1,1,0,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => 2
[1,0,1,1,0,0] => [1,3,2] => 2
[1,1,0,0,1,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => 2
[1,1,1,0,0,0] => [3,1,2] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [2,4,1,3] => 2
[1,1,1,0,0,0,1,0] => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => 2
[1,1,1,0,1,0,0,0] => [3,4,1,2] => 3
[1,1,1,1,0,0,0,0] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 4
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 3
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 4
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 4
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => 4
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => 2
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 4
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 4
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 4
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 4
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 3
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 3
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => 4
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 5
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => 5
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 5
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 5
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 5
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => 4
>>> Load all 195 entries. <<<
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,6,4] => 4
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,2,6,3,4] => 3
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,2,3,4] => 5
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 5
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 5
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 4
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => 5
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,4,5] => 4
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 5
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 5
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => 5
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,3,5] => 4
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,3,4,6] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,3,6,4] => 4
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,3,4] => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,3,4,5] => 3
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 5
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => 5
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => 4
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 5
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,4,5] => 4
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 5
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => 5
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 5
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,1,5] => 4
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,1,4,6] => 4
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,1,6,4] => 4
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,1,4] => 5
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,1,4,5] => 3
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,1,3,5,6] => 5
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,1,3,6,5] => 5
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,1,5,3,6] => 3
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,1,5,6,3] => 5
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,1,6,3,5] => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,1,3,6] => 3
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,1,6,3] => 3
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,1,3] => 5
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,1,3,5] => 4
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,1,3,4,6] => 4
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,1,3,6,4] => 4
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,1,6,3,4] => 3
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,1,3,4] => 5
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,1,3,4,5] => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,2,4,5,6] => 5
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2,4,6,5] => 5
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,1,2,5,4,6] => 4
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,1,2,5,6,4] => 5
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,1,2,6,4,5] => 4
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => 5
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,1,4,2,6,5] => 5
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,1,4,5,2,6] => 2
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => 5
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,1,4,6,2,5] => 4
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,1,5,2,4,6] => 4
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,1,5,2,6,4] => 4
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,1,5,6,2,4] => 5
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,1,6,2,4,5] => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,1,2,5,6] => 5
[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,1,2,6,5] => 5
[1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,1,5,2,6] => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => 5
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,1,6,2,5] => 2
[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,1,2,6] => 2
[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => 5
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => 4
[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,1,2,4,6] => 4
[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => 4
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,1,6,2,4] => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => 5
[1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,2,3,5,6] => 5
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,1,2,3,6,5] => 5
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,1,2,5,3,6] => 3
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => 5
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,1,2,6,3,5] => 3
[1,1,1,1,0,0,1,0,0,0,1,0] => [4,1,5,2,3,6] => 3
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,1,5,2,6,3] => 3
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,1,5,6,2,3] => 5
[1,1,1,1,0,0,1,1,0,0,0,0] => [4,1,6,2,3,5] => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,1,2,3,6] => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => 3
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,1,6,2,3] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => 5
[1,1,1,1,0,1,1,0,0,0,0,0] => [4,6,1,2,3,5] => 4
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6] => 4
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => 4
[1,1,1,1,1,0,0,0,1,0,0,0] => [5,1,2,6,3,4] => 3
[1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => 2
[1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,1,2,3,4] => 5
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The greater neighbor of the maximum.
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the smallest path leaf label of the binary tree associated to a permutation (St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.), see also [3].
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.