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A115728
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Number of subpartitions of partitions in Abramowitz and Stegun order.
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26
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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, 13, 10, 14, 13, 10, 12, 11, 7, 8, 13, 15, 16, 14, 19, 17, 16, 16, 19, 16, 14, 15, 13, 8, 9, 15, 18, 19, 18, 24, 21, 15, 23, 22, 26, 21, 19, 22, 23, 24, 19, 15, 18, 18, 15, 9, 10, 17, 21, 22, 22, 29
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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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).
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
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EXAMPLE
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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
3, 3
4, 5, 4
5, 7, 6, 7, 5
6, 9, 9, 10, 9, 9, 6
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PROG
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(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, Jul 03 2006
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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