A179094 Disorder number of the n X n grid graph.

0, 5, 23, 61, 119, 213, 335, 509, 719, 997, 1319, 1725, 2183, 2741, 3359, 4093, 4895

Offset: 1

Author

Thomas Young, Jun 29 2010

Keywords

• nonn
• more

Comments

Old name: Fill an n X n array with various permutations of the integers 1, 2, 3, 4... n^2. Define the organization number of the n X n array to be the following: Start at 1, count the rectilinear steps to reach 2, then the rectilinear steps to reach 3, etc. Add them up. The array that has the maximum organization number would be the "most disorganized." This sequence is the sequence showing the most disorganized number for n X n arrays starting at 1 X 1.

Similar to sequence A047838.

My computer program worked as follows:

a) generate a permutation

b) place the permutation into the array

c) calculate the array position (row, column) of each integer

d) sort the integers into another array preserving row and column

e) travel the new array from 1..n^2 and summing the absolute value of the differences between the rows of consecutive integers and summing the absolute value of the differences of the columns of consecutive integers. The organization number is the sum of the two sums.

For instance, with the permutation 8, 3, 6, 5, 9, 1, 2, 7, 4 place the integers into a 3 X 3 array as such:

8 3 6

5 9 1

2 7 4

(Notice the next integer is a knight's move away. This is not the only sequence that will give an organization number of 23, but this is why I wonder if the sequence is the same as A098499.)

Then sort the integers preserving their row and column:

number, row, column

1, 2, 3

2, 3, 1

3, 1, 2

4, 3, 3

5, 2, 1

6, 1, 3

7, 3, 2

8, 1, 1

9, 2, 2

Traveling from 1 to 9, the differences in the row numbers are 1, 2, 2, 1, 1, 2, 2, 1 (a sum of 12) and the differences in the column numbers are 2, 1, 1, 2, 2, 1, 1, 1 (a sum of 11) therefore the organization number is 23.

This is basically a traveling salesman variant. - D. S. McNeil, Aug 26 2010

Links

• Sela Fried, On a conjecture of McNeil, arXiv:2208.03788 [math.CO], 2023.
• Eric Weisstein's World of Mathematics, Disorder Number.
• Eric Weisstein's World of Mathematics, Grid Graph

Crossrefs

Cf. A047838, A098499.

Formula

A possible formula: a(n) = 0 for n=1, n^3-n-1 for odd n > 1, n^3-3 for even n? - D. S. McNeil, Aug 26 2010

Let b(n) correspond to McNeil's formula. Then b(n) <= a(n) <= b(n) + 1 (see link). - Sela Fried, Nov 28 2023

Empirical G.f.: x^2*(5+13*x+10*x^2-6*x^3+x^4+x^5)/((1-x)^4*(1+x)^2). - Colin Barker, Mar 29 2012

Extensions

a(3) corrected and a(4)-a(17) computed by D. S. McNeil, Aug 26 2010. D. S. McNeil also finds that a(19)=6839, a(21)=9239, a(23)=12143.

Edited by N. J. A. Sloane, Aug 26 2010

Typo in formula corrected by D. S. McNeil, Aug 26 2010

Equivalent simpler description from Eric W. Weisstein, Oct 08 2024