Help deriving integer sequence formula

[ktoz]

Hi

I'm playing around with partitions and have come up with an integer sequence representing the maximum number of partitions of various "widths" that display the following properties:

- min values in partition are equal
- max values in partition are equal
- partitions contain equal number of members
- sum of members is equal

For example, given:

min = 1
max = 6
count = 4
sum = 14

There are only two partitions that satisfy the constraints

{1,3,4,6}
{1,2,5,6}

Using a brute force algorithm, I came up with the following maximums for width = {1, 2, 3, 4 ..., 24}

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910, 1667, 2934, 5448, 9686, 18084

My algorithm breaks at 25 due to the huge memory trequirements needed to sample every possible combination. I plugged it into http://www.research.att.com/~njas/sequences/" , but no luck.

With a little tweaking, the series seems like it might have some sort of partial relationship with the Fibonacci and Lucas series, but I haven't been able to come up with anything concrete.

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910
 ,  ,  ,  ,  ,  , 1, 2, 3, 5, 8,  13, 21, 34, 55,  89, 144, 233, 377 	(fib)
 --------------------------------------------------------------------
		  1, 1, 2, 3, 4,  7,  11, 24, 39,  86, 145, 293, 533	(partial lucas)

Anyone see the pattern? Or perhaps someone with Mathematica 7 could plug the series into the series calculator and come up with the formula?

Thanks for any help

 

[ramsey2879]

Hi

I'm playing around with partitions and have come up with an integer sequence representing the maximum number of partitions of various "widths" that display the following properties:

- min values in partition are equal
- max values in partition are equal
- partitions contain equal number of members
- sum of members is equal

For example, given:

min = 1
max = 6
count = 4
sum = 14

There are only two partitions that satisfy the constraints

{1,3,4,6}
{1,2,5,6}

Using a brute force algorithm, I came up with the following maximums for width = {1, 2, 3, 4 ..., 24}

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910, 1667, 2934, 5448, 9686, 18084

My algorithm breaks at 25 due to the huge memory trequirements needed to sample every possible combination. I plugged it into http://www.research.att.com/~njas/sequences/" , but no luck.

With a little tweaking, the series seems like it might have some sort of partial relationship with the Fibonacci and Lucas series, but I haven't been able to come up with anything concrete.

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910
 ,  ,  ,  ,  ,  , 1, 2, 3, 5, 8,  13, 21, 34, 55,  89, 144, 233, 377 	(fib)
 --------------------------------------------------------------------
		  1, 1, 2, 3, 4,  7,  11, 24, 39,  86, 145, 293, 533	(partial lucas)

Anyone see the pattern? Or perhaps someone with Mathematica 7 could plug the series into the series calculator and come up with the formula?

Thanks for any help

Could you define "width"? You give 4 variables for widths but you give a sequence for widths = [1,2,3,...} so I don't understand what you mean.