Identifier
- St000887: Permutations ⟶ ℤ
Values
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 3
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
[1,5,2,3,4] => 3
[1,5,2,4,3] => 2
[1,5,3,2,4] => 2
[1,5,3,4,2] => 3
[1,5,4,2,3] => 2
[1,5,4,3,2] => 1
[2,1,3,4,5] => 3
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 3
[2,1,5,3,4] => 3
[2,1,5,4,3] => 1
[2,3,1,4,5] => 2
[2,3,1,5,4] => 3
[2,3,4,1,5] => 3
[2,3,4,5,1] => 4
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[2,4,1,3,5] => 1
[2,4,1,5,3] => 2
[2,4,3,1,5] => 2
[2,4,3,5,1] => 2
[2,4,5,1,3] => 2
[2,4,5,3,1] => 2
[2,5,1,3,4] => 2
[2,5,1,4,3] => 2
[2,5,3,1,4] => 1
[2,5,3,4,1] => 2
[2,5,4,1,3] => 2
[2,5,4,3,1] => 2
[3,1,2,4,5] => 2
[3,1,2,5,4] => 3
[3,1,4,2,5] => 1
[3,1,4,5,2] => 2
[3,1,5,2,4] => 2
[3,1,5,4,2] => 2
[3,2,1,4,5] => 3
[3,2,1,5,4] => 1
[3,2,4,1,5] => 2
[3,2,4,5,1] => 2
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 2
[3,4,1,5,2] => 2
[3,4,2,1,5] => 2
[3,4,2,5,1] => 2
[3,4,5,1,2] => 3
[3,4,5,2,1] => 3
[3,5,1,2,4] => 2
[3,5,1,4,2] => 1
[3,5,2,1,4] => 2
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Description
The maximal number of nonzero entries on a diagonal of a permutation matrix.
For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix},$$ and the entries corresponding to $\pi_2=1$, $\pi_3=2$ and $\pi_5=4$ are all on the fourth diagonal from the right.
In other words, this is $\max_k \lvert\{i: \pi_i-i = k\}\rvert$
For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix},$$ and the entries corresponding to $\pi_2=1$, $\pi_3=2$ and $\pi_5=4$ are all on the fourth diagonal from the right.
In other words, this is $\max_k \lvert\{i: \pi_i-i = k\}\rvert$
References
[1] Nordh, G. A note on X-rays of permutations and a problem of Brualdi and Fritscher arXiv:1707.03928
Code
def statistic(pi):
P = Permutation(pi).to_matrix()
n = P.nrows()
return max(sum(P[n-k+j][j] for j in range(max(0, k-n), min(k, n)))
for k in range(1, 2*n))
Created
Jul 14, 2017 at 08:49 by Martin Rubey
Updated
Jul 17, 2017 at 07:31 by Martin Rubey
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