Maximize/Minimize It

Let x1,x2,,x19,x20x_1,x_2,\cdots,x_{19},x_{20} be permutations of numbers from 1 to 20. Then what will be the maximum value of the summation x1x2+x2x3++x19x20+x20x1\large x_1x_2+x_2x_3+\cdots+x_{19}x_{20}+x_{20}x_1 At which values of x1,x2,,x19,x20x_1,x_2,\cdots,x_{19},x_{20} will that maximum occur? And when will minimum occur?

Note by Vilakshan Gupta
1 year, 11 months ago

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I looked at this a little but I haven't done anything substantive. It seems like it should be possible to show that the minimum is obtained by alternating large and small numbers in a smart way.

E.g. for n=4n=4 you get a minimum when you do something like 3142.3142.

For n=6n=6 you do 351624.351624.

For n=8n=8 you do 53718264.53718264.

For n=10n=10 you do 57391(10)2846.57391(10)2846. And so on.

Inductively, to get from one level to the next, it looks like you add two to the numbers larger than n/2,n/2, and then on the outside you stick n/2n/2 and n/2+1n/2+1 in a smart way.

I don't have any kind of proof though, and I haven't thought about the maximum yet.

Patrick Corn - 1 year, 11 months ago

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For the maximum for n=Nn = N, going from NN down to 11 while alternating from one side of the sequence to the other might be optimal. For n=4n = 4 this would be 42134213, yielding a product-sum of 2525, (compared to your minimum of 2121), and for n=6n = 6 it would be 642135642135, yielding a product-sum of 8282, (compared to your minimum of 5858).

Brian Charlesworth - 1 year, 11 months ago

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Here's one idea I tried: for it to be a minimum (resp. maximum), the result has to be smaller (resp. bigger) than what you get after doing a transposition. For say x2,x3,x_2,x_3, this ends up being the statement that (x1x4)(x2x3)<0(x_1-x_4)(x_2-x_3) < 0 (resp. >0>0). And so on for other pairs of adjacent indices. Your examples satisfy this criterion.

There is another condition you get if you transpose x2,x4x_2,x_4 (say), and then you can look at a 33-cycle too. Taken together, those give you pretty strong conditions on the permutation, but I don't know if they determine it uniquely.

Patrick Corn - 1 year, 11 months ago

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Please help me with this anyone. @Brian Charlesworth , @Chew-Seong Cheong , @Anirudh Sreekumar , @Sharky Kesa anyone?

Vilakshan Gupta - 1 year, 11 months ago

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again

Ayush Jain - 1 year, 11 months ago

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wc

Ayush Jain - 1 year, 11 months ago

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explain this

Ayush Jain - 1 year, 11 months ago

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