Identifier
Values
[2] => [[1,2]] => [1,2] => [1,2] => 1
[1,1] => [[1],[2]] => [2,1] => [2,1] => 1
[3] => [[1,2,3]] => [1,2,3] => [1,3,2] => 1
[2,1] => [[1,2],[3]] => [3,1,2] => [3,1,2] => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,4,3,2] => 1
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [4,1,3,2] => 2
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 1
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [4,3,1,2] => 3
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [5,1,4,3,2] => 2
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [4,5,1,3,2] => 2
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [5,4,1,3,2] => 3
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [5,3,4,1,2] => 3
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [5,4,3,1,2] => 4
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,6,5,4,3,2] => 1
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [6,1,5,4,3,2] => 2
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [5,6,1,4,3,2] => 2
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [6,5,1,4,3,2] => 3
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => 1
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [6,4,5,1,3,2] => 6
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [6,5,4,1,3,2] => 4
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 1
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [6,5,3,4,1,2] => 6
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [6,5,4,3,1,2] => 5
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
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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
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.