Partitions Based On Permutations

[Leroy Quet]

I just submitted this sequence:

%I A000001
%S A000001
1,0,0,1,1,0,0,0,0,1,2,0,2,1,0,0,0,0,0,1,3
%N A000001 a(n) = the total number of
permutations (m(1),m(2),m(3)...m(j)) of
(1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3)
+ ...+j*m(j), where j is over all positive
integers.
%C A000001 Does every integer greater than some
positive integer N have at least one such
representation?
%e A000001 21 has a(21)=3 such representations:
21 = 1*4 + 2*3 + 3*1 + 4*2 = 1*4 + 2*2 + 3*3 +
4*1 = 1*3 + 2*4 + 3*2 + 4*1.
Not all representations of an integer n need to
necessarily have the same j. For example, 91 =
1*1 + 2*2 + 3*3 + 4*4 + 5*5 + 6*6 (j=6). And 91
also equals 1*7 + 2*4 + 3*5 + 4*3 + 5*6 + 6*2 +
7*1 (j=7).
%O A000001 1
%K A000001 ,more,nonn,

I know I haven't given the best definition.
Hopefully what I mean is clear. (The examples
should help a little.)

Does every integer greater than some integer
(such as, say, 20) have such a representation?

It seems very likely that the answer is yes. For
a given j, the least  possible n is j^3/6 + j^2/2
+ j/3. The greatest possible n is j^3/3 + j^2/2
+j/6 (the sum of the first j squares).
The difference between these extremes, plus 1, is
j^3/6 - j/6 + 1, if I did my math right.
But the number of permutations for a given j is
j!, obviously.
So the chance that any particular integer between
the maximum n and the minimum n, for a given j,
does not have a representation gets low pretty
quick.

Thanks,
Leroy Quet



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