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For a partition P = [p_1,...,p_n] with the p_i in increasing 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=0}^..j} b(i,k) for 0<=j<=p_{i+1}; and the total number of subpartitions is sum_{k=1}^{..p_n} b(n,k).
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Partition 5 in A&S order is [2,1]; it has 5 subpartitions: [], [1], [2], [1^2] and [2,1] itself.
[], [1], [2], [1^2] and [2,1] itself.
(PARI) /* Expects input as vector in increasing order - e.g. [1, 1, 2, 3] */
(PARI) /* Expects input as vector in increasing order - e.g. [1, 1, 2, 3] */ subpart(p)=local(i, j, v, n); n=matsize(p)[2]; if(n==0, 1, v=vector(p[n]+1); v[1] =1; for(i=1, n, for(j=1, p[i], v[j+1]+=v[j])); for(j=1, p[n], v[j+1]+=v[j]); v[p[n ]+1])
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(PARI) /* Given Partition p(), Find Subpartitions s(): */ {s(n)=polcoeff(x^n-sum(k=0, n-1, s(k)*x^k*(1-x+x*O(x^n))^p(k)), n)} - _\\ _Paul D. Hanna_, Jul 03 2006
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