On graph paper I render the partition 6+4+2+1+1 as a vertical stack of 6 squares, next to a stack of 4 squares, etc. for a total of 5 vertical stacks (the last two comprising just a single square): x x x x x x x x x x x x x x The leftmost stack is at x=0 and the bottom row is y=0. The formula I gave has the sizes of the squares rescaled so the area under the curve is independent of N. -Veit
On Feb 11, 2019, at 4:49 PM, Allan Wechsler <acwacw@gmail.com> wrote:
It's not clear to me exactly what is subsumed under "rescaling". Do you scale horizontally, so that the rescaled partitions look like they all have the same number of parts? Do you scale vertically, so that they look like they all have the same largest part? Both?
On Mon, Feb 11, 2019 at 2:08 PM Veit Elser <ve10@cornell.edu> wrote:
If you uniformly sample from the partitions of an integer N, and interpret the sequence of parts, sorted largest to smallest, as a (decreasing) function y(x), then in the limit of large N the y(x) of the “typical" partition satisfies (after rescaling)
(e^x-1)(e^y-1) = 1
Does this curve have a name?
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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