0,2

subpart([n^k]) = C(n+k,k); subpart([1,2,3,...,n]) = C_n = A000108(n). The b(i,j) defined in the formula for sequences [1,2,3,...] form A009766.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

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).

For a partition P = {p(n)}, the number of subpartitions {s(n)} of P can be determined by the g.f.: 1/(1-x) = Sum_{n>=0} s(n)*x^n*(1-x)^p(n). - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 03 2006

Partition 5 in A&S order is [2,1]; it has 5 subpartitions:

[], [1], [2], [1^2], and [2,1] itself.

(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])

(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 (pauldhanna(AT)juno.com), Jul 03 2006

Cf. A115729, A036036, A000108, A009766, A007318.

Sequence in context: A094727 A089308 A115729 this_sequence A026354 A078338 A007306

Adjacent sequences: A115725 A115726 A115727 this_sequence A115729 A115730 A115731

nonn

Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 11 2006