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

On Apr 6, 11:30 am, "joeship...@xxxxxxx" <JoeShip...@xxxxxxx> wrote:
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.

To clarify what I just wrote -- in the above example, you have to pad
the shorter multiset with 0's until it is the same length as the
longer one to make the "0th moment" (cardinality) the same -- it is
the multiset {0,3,3} that has the same mean and standard deviation as
{1,1,4}, not the multiset {3,3}.

In that example, there is an obvious symmetry -- if you normalize so
the mean is 0 then the multisets become {-2,1,1} and {-1,-1,2} which
have the same moments of even order and odd moments which have
opposite signs.

Another example with the same symmetry is {0,2,2,2} and {3,1,1,1}. The
smallest example which doesn't have this symmetry is {1,1,1,1,5}--
{0,0,2,3,4}. You can construct a similar example from any pythagorean
triple, thus the pythagorean triple (8,15,17) gives rise to the two
multisets {0,0,0,0,0,0,0,0,2,2,2,8,15}--{1,1,1,1,1,1,1,1,1,1,1,1,17}
which have the same cardinality, sum, and sum-of-squares.

But I want to find small examples which also have the same sum-of-
cubes and sum-of4th-powers.

-- Joe Shipman

.

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