Semi Group and interval partitions

Semi group is a group except that there is no inverse. Remember it has a zero, closure and associativity. One of the homework set problem is to prove that S = {n,r1x1+r2*x2+ rn*xn = n} forms a semigroup.
I was also wondering about the equivalence of partition in subsets to partition in intervals. We all know that the total number of partitions p(n) of set [n] is given by BellNumber(n) but what about the interval [n]. Dr. Clark started with a k interval as (1+..+x1)+(x1+1+...+x1+x2)+(x2+1...x2+x3)+ ... (x(k-1)+1....+ x(k)=n)
Note that once you write this way the size of each partition is x1,x2,x3,...xk or in other words
x1+x2+..xk = n which is just integer composition and if each is non zero then the total number of solution is Binomial(n-1,k-1) and Sum(Binomial(n-1,k-1),k =1..n) = 2^(n-1). Thus number of intervals on [n] = 2^(n-1).
Verification
n= 1
2^(1-1) = 2^0 =1
which makes sense
n =2
2^(2-1) = 2
unitary elements {1},{2}
and {1,2}

n = 3
2^(3-1) = 4
unitary elements {1},{2},{3}
Two block
{1}{2,3},
{1,2},{3}
Single block {1,2,3}