OFFSET
0,2
COMMENTS
subpart([n^k]) = C(n+k,k); subpart([n,n-1,n-2,...,1]) = C_n = A000108(n).
FORMULA
For a partition P = [p_1,...,p_n] with the p_i in decreasing order, define b(i,j) to be the number of subpartitions of [p_1,...,p_i] with the i-th part = j (b(i,0) is subpartitions with less than i parts). Then b(1,j)=1 for j<=p_1, b(i+1,j) = Sum_{k=j}^{p_i} b(i,k) for 0<=k<=p_{i+1}; and the total number of subpartitions is sum_{k=1}^{p_n} b(n,k).
EXAMPLE
Partition 5 in Mathematica order is [2,1]; it has 5
subpartitions: [], [1], [2], [1^2] and [2,1] itself.
PROG
(PARI) /* Expects input as vector in decreasing order - e.g. [3, 2, 1, 1] */ subpart2(p)=local(i, j, v, n, k); n=matsize(p)[2]; if(n==0, 1, v=vector(p[1]+1, i, 1); for(i=1, n, k=p[i]; for(j=1, k, v[k+1-j]+=v[k+2-j])); v[1])
CROSSREFS
KEYWORD
nonn
AUTHOR
Franklin T. Adams-Watters, Mar 11 2006
STATUS
approved