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
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
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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you mean rescaling x and y by a factor of sqrt(n), right? - Cris
On Feb 11, 2019, at 12:07 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
I’m pretty sure I’ve seen the curve before, in work on random partitions from the late 20th century. But I don’t think the curve was given a name. Jim On Monday, February 11, 2019, Veit Elser <ve10@cornell.edu> wrote:
Right. You can get the numerical factor by finding the area under the curve.
Nobody has seen this curve before?
On Feb 11, 2019, at 8:23 PM, Cris Moore <moore@santafe.edu> wrote:
you mean rescaling x and y by a factor of sqrt(n), right? - Cris
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Take a look at Knuth, volume 4A, page 402, Fig 50, Temperley's curve Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Feb 11, 2019 at 9:18 PM James Propp <jamespropp@gmail.com> wrote:
I’m pretty sure I’ve seen the curve before, in work on random partitions from the late 20th century. But I don’t think the curve was given a name.
Jim
On Monday, February 11, 2019, Veit Elser <ve10@cornell.edu> wrote:
Right. You can get the numerical factor by finding the area under the curve.
Nobody has seen this curve before?
On Feb 11, 2019, at 8:23 PM, Cris Moore <moore@santafe.edu> wrote:
you mean rescaling x and y by a factor of sqrt(n), right? - Cris
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participants (5)
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Allan Wechsler -
Cris Moore -
James Propp -
Neil Sloane -
Veit Elser