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A115729
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Number of subpartitions of partitions in Mathematica order.
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10
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| subpart([n^k]) = C(n+k,k); subpart([n,n-1,n-2,...,1]) = C_n = A000108(n).
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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).
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EXAMPLE
| Partition 5 in Mathematica order is [2,1]; it has 5
subpartitions: [], [1], [2], [1^2] and [2,1] itself.
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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])
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CROSSREFS
| Cf. A115728, A080577, A000108, A007318.
Sequence in context: A120244 A094727 A089308 * A115728 A188553 A026354
Adjacent sequences: A115726 A115727 A115728 * A115730 A115731 A115732
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KEYWORD
| nonn
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AUTHOR
| Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 11 2006
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