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 A115729 Number of subpartitions of partitions in Mathematica order. 12
 1, 2, 3, 3, 4, 5, 4, 5, 7, 6, 7, 5, 6, 9, 9, 10, 9, 9, 6, 7, 11, 12, 10, 13, 14, 10, 13, 12, 11, 7, 8, 13, 15, 14, 16, 19, 16, 16, 17, 19, 14, 16, 15, 13, 8, 9, 15, 18, 18, 15, 19, 24, 23, 22, 19, 21, 26, 22, 23, 15, 21, 24, 18, 19, 18, 15, 9, 10, 17, 21, 22, 20, 22 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS subpart([n^k]) = C(n+k,k); subpart([n,n-1,n-2,...,1]) = C_n = A000108(n). LINKS 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 Cf. A115728, A080577, A000108, A007318. Sequence in context: A089308 A305579 A321440 * A115728 A188553 A026354 Adjacent sequences:  A115726 A115727 A115728 * A115730 A115731 A115732 KEYWORD nonn AUTHOR Franklin T. Adams-Watters, Mar 11 2006 STATUS approved

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Last modified January 18 13:30 EST 2020. Contains 331007 sequences. (Running on oeis4.)