Re: Multisets of integers with equal sums, sums-of-squares, etc.

"joeshipman@xxxxxxx" <JoeShipman@xxxxxxx> writes:

The multisets {3,3} and {1,1,4} have equal sums and equal sums-of-
squares. What's the smallest example of two multisets of integers
which have the same sums-of-nth-powers for n=1,2,3? What's the
smallest example for n=1,2,3,4?

I am interested in this because multisets with equal sums and sums-of-
squares have the same mean and standard deviation. Sampling with
replacement from these multisets ought to converge to the same thing
by the Central Limit Theorem, but I want to illustrate how the rate of
convergence depends on higher moments, using examples that are
numerically simple.

If you're interested in equal mean and standard deviation you also want
the sum of 0'th powers, i.e. the cardinalities, to be the same.

The smallest multisets of integers with sum 0 that are not symmetric about 0
are {-1,-1,2} and {-2,1,1}. Thus these have the same first moment, and clearly
they have the same zeroth and second moments.

I'm not sure if it's "smallest" in any sense, but {-4,-4,1,1,1,5}
is an example of a multiset with sum and third moment 0, and therefore
this and {-5,-1,-1,-1,4,4} have the same zeroth to fourth moments.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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