[seqfan] A problem related to A027336.

[L. Edson Jeffery]

The following contains a problem and a conjecture related (I think) to
A027336:

Let n be a positive integer. Let A(n) be the set of all compositions of n
with general j-th element

[A(n)]_j = <a(j,1), a(j,2),...,a(j,k_j)>

with the restrictions

(1)  a(j,1) <= a(j,2) <= ... <= a(j,k_j)

on ordering of the parts, where k_j is the number of parts, 1 <= k_j <= n,
1 <= j <= p(n), and p(n) = A000041(n) is the number of partitions of n.

Problem: Let m(n) denote the number of elements of A(n) with the additional
restrictions

    /  k_j = 1
(2)-|  or:
    \  k_j>2 and a(j, k_j - 1) = a(j, k_j - 2);

then what is the sequence {m(n)}?

Conjecture: For all v <= 0, let p(v) = 0. For all positive integers n, m(n)
= p(n) - p(n-2) = A027336(n).

Can someone prove (or disprove) this conjecture?

The ascending order of the parts combined with the restrictions (2) does
not seem to match any of the definitions listed in the %N or %C sections or
in the references in A027336.

Any help with this problem will be greatly appreciated.

Ed Jeffery