Partitions Based On Permutations

[Augustine Munagi]

Hi Leroy,
I have not looked carefully at the problem but it may be characterized
directly by a class of distinct partitions of n. Taking the conjugate
of the partition n = 1*m(1) + 2*m(2) + 3*m(3) gives a partition of n
into distinct parts with further restriction. So an admissible
representation corresponds to a partition of n, say n = d1+d2+....+dj,
with d1<d2<...<dj, such that the sequence of adjacent differences
(d1,d2-d1,d3-d2,....,dj-d_(j-1)) is a permutation of {1,2,...,j}.

Thanks,

Augustine



On 12/4/07, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
> 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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