[seqfan] A208250 and growth of the expected largest bucket
hv at crypt.org
hv at crypt.org
Wed Feb 20 09:09:48 CET 2013
Could someone help me consider how to calculate more values of A208250,
or to approximate them? I'd like to find values for rather larger n,
but I'm struggling to think of an approach that will make it practical
to calculate for n substantially greater than the 19 already listed.
Working a(4) by hand, for example, I get:
(4,0,0,0) has largest bucket 4; can appear 4 ways; can be constructed in
only one order; contribution is 4 * 4 * 1 => 16.
(3,1,0,0) => largest bucket 3; 12 ways; 4 orders => 144.
(2,2,0,0) => 2 * 6 * 6 => 72.
(2,1,1,0) => 2 * 12 * 12 => 288.
(1,1,1,1) => 1 * 1 * 24 => 24.
total 16+144+72+288+24 = 544
.. but an approach iterating over all partitions of n is not going to scale
very far.
Motivation: for a programming-related issue I need to get a clearer picture
of how the expected size of largest bucket b(n) = A208250(n) / n^n will grow,
particularly for n = 2^m. From the current values I get:
b(2^0) = 1
b(2^1) = 1.5
b(2^2) = 2.125
b(2^3) = 2.597 (approx)
b(2^4) = 3.078 (approx)
Hugo
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