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%I #21 Jan 22 2021 21:24:22
%S 1,2,3,3,4,5,4,5,7,6,7,5,6,9,9,10,9,9,6,7,11,12,13,10,14,13,10,12,11,
%T 7,8,13,15,16,14,19,17,16,16,19,16,14,15,13,8,9,15,18,19,18,24,21,15,
%U 23,22,26,21,19,22,23,24,19,15,18,18,15,9,10,17,21,22,22,29
%N Number of subpartitions of partitions in Abramowitz and Stegun order.
%C 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.
%C Row sums are A297388. Row lengths are A000041. - _Geoffrey Critzer_, Jan 10 2021
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F 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).
%F 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_, Jul 03 2006
%e Partition 5 in A&S order is [2,1]; it has 5 subpartitions: [], [1], [2], [1^2] and [2,1] itself.
%e 1
%e 2
%e 3, 3
%e 4, 5, 4
%e 5, 7, 6, 7, 5
%e 6, 9, 9, 10, 9, 9, 6
%o (PARI) /* Expects input as vector in increasing order - e.g. [1,1,2,3] */
%o 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])
%o (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
%Y Cf. A115729, A036036, A000108, A009766, A007318, A297388, A000041.
%K nonn
%O 0,2
%A _Franklin T. Adams-Watters_, Mar 11 2006
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