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A115728
Number of subpartitions of partitions in Abramowitz and Stegun order.
26
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
OFFSET
0,2
COMMENTS
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.
Row sums are A297388. Row lengths are A000041. - Geoffrey Critzer, Jan 10 2021
LINKS
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].
FORMULA
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
EXAMPLE
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
PROG
(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
KEYWORD
nonn
AUTHOR
STATUS
approved