Identifier
- St000342: Permutations ⟶ ℤ (values match St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.)
Values
[1] => 1
[1,2] => 5
[2,1] => 4
[1,2,3] => 14
[1,3,2] => 13
[2,1,3] => 13
[2,3,1] => 11
[3,1,2] => 11
[3,2,1] => 10
[1,2,3,4] => 30
[1,2,4,3] => 29
[1,3,2,4] => 29
[1,3,4,2] => 27
[1,4,2,3] => 27
[1,4,3,2] => 26
[2,1,3,4] => 29
[2,1,4,3] => 28
[2,3,1,4] => 27
[2,3,4,1] => 24
[2,4,1,3] => 25
[2,4,3,1] => 23
[3,1,2,4] => 27
[3,1,4,2] => 25
[3,2,1,4] => 26
[3,2,4,1] => 23
[3,4,1,2] => 22
[3,4,2,1] => 21
[4,1,2,3] => 24
[4,1,3,2] => 23
[4,2,1,3] => 23
[4,2,3,1] => 21
[4,3,1,2] => 21
[4,3,2,1] => 20
[1,2,3,4,5] => 55
[1,2,3,5,4] => 54
[1,2,4,3,5] => 54
[1,2,4,5,3] => 52
[1,2,5,3,4] => 52
[1,2,5,4,3] => 51
[1,3,2,4,5] => 54
[1,3,2,5,4] => 53
[1,3,4,2,5] => 52
[1,3,4,5,2] => 49
[1,3,5,2,4] => 50
[1,3,5,4,2] => 48
[1,4,2,3,5] => 52
[1,4,2,5,3] => 50
[1,4,3,2,5] => 51
[1,4,3,5,2] => 48
[1,4,5,2,3] => 47
[1,4,5,3,2] => 46
[1,5,2,3,4] => 49
[1,5,2,4,3] => 48
[1,5,3,2,4] => 48
[1,5,3,4,2] => 46
[1,5,4,2,3] => 46
[1,5,4,3,2] => 45
[2,1,3,4,5] => 54
[2,1,3,5,4] => 53
[2,1,4,3,5] => 53
[2,1,4,5,3] => 51
[2,1,5,3,4] => 51
[2,1,5,4,3] => 50
[2,3,1,4,5] => 52
[2,3,1,5,4] => 51
[2,3,4,1,5] => 49
[2,3,4,5,1] => 45
[2,3,5,1,4] => 47
[2,3,5,4,1] => 44
[2,4,1,3,5] => 50
[2,4,1,5,3] => 48
[2,4,3,1,5] => 48
[2,4,3,5,1] => 44
[2,4,5,1,3] => 44
[2,4,5,3,1] => 42
[2,5,1,3,4] => 47
[2,5,1,4,3] => 46
[2,5,3,1,4] => 45
[2,5,3,4,1] => 42
[2,5,4,1,3] => 43
[2,5,4,3,1] => 41
[3,1,2,4,5] => 52
[3,1,2,5,4] => 51
[3,1,4,2,5] => 50
[3,1,4,5,2] => 47
[3,1,5,2,4] => 48
[3,1,5,4,2] => 46
[3,2,1,4,5] => 51
[3,2,1,5,4] => 50
[3,2,4,1,5] => 48
[3,2,4,5,1] => 44
[3,2,5,1,4] => 46
[3,2,5,4,1] => 43
[3,4,1,2,5] => 47
[3,4,1,5,2] => 44
[3,4,2,1,5] => 46
[3,4,2,5,1] => 42
[3,4,5,1,2] => 40
[3,4,5,2,1] => 39
[3,5,1,2,4] => 44
[3,5,1,4,2] => 42
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Description
The cosine of a permutation.
For a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this is given by $\sum_{i=1}^n (i\pi_i)$.
The name comes from the observation that this equals $\frac{n(n+1)(2n+1)}{6}\cos(\theta)$ where $\theta$ is the angle between the vector $(\pi_1,\ldots,\pi_n)$ and the vector $(1,\ldots,n)$, see [1].
For a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this is given by $\sum_{i=1}^n (i\pi_i)$.
The name comes from the observation that this equals $\frac{n(n+1)(2n+1)}{6}\cos(\theta)$ where $\theta$ is the angle between the vector $(\pi_1,\ldots,\pi_n)$ and the vector $(1,\ldots,n)$, see [1].
References
[1] Sack, J., Úlfarsson, H. Refined inversion statistics on permutations MathSciNet:2880660 arXiv:1106.1995
[2] a(n) = the total number of permutations (m(1),m(2),m(3)...m(j)) of (1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where j is over all positive integers. OEIS:A135298
[2] a(n) = the total number of permutations (m(1),m(2),m(3)...m(j)) of (1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where j is over all positive integers. OEIS:A135298
Code
def statistic(pi):
return sum( (i+1)*val for i,val in enumerate(pi) )
Created
Dec 23, 2015 at 08:44 by Christian Stump
Updated
May 10, 2019 at 17:24 by Henning Ulfarsson
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