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 Sloan's, 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. Code (Text): 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.