Identifier
Values
[1,0] => [2,1] => 1
[1,0,1,0] => [3,1,2] => 2
[1,1,0,0] => [2,3,1] => 2
[1,0,1,0,1,0] => [4,1,2,3] => 2
[1,0,1,1,0,0] => [3,1,4,2] => 2
[1,1,0,0,1,0] => [2,4,1,3] => 2
[1,1,0,1,0,0] => [4,3,1,2] => 3
[1,1,1,0,0,0] => [2,3,4,1] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 4
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 4
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 3
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 3
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 8
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 6
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 6
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 4
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 8
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 4
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 6
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 4
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 6
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 7
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 4
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 4
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 4
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 7
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 4
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 8
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 4
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 8
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 12
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 4
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Description
The number of permutations with the same antidiagonal sums.
The X-ray of a permutation $\pi$ is the vector of the sums of the antidiagonals of the permutation matrix of $\pi$, read from left to right. For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is
$$\left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right),$$
so its X-ray is $(0, 1, 1, 1, 0, 0, 0, 2, 0)$.
This statistic records the number of permutations having the same X-ray as the given permutation. In [1] this is called the degeneracy of the X-ray of the permutation.
By [prop.1, 1], the number of different X-rays of permutations of size $n$ equals the number of nondecreasing differences of permutations of size $n$, [2].
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.