Search: a316222 -id:a316222
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A316223
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Number of subset-sum triangles with composite a subset-sum of the integer partition with Heinz number n.
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+10
6
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0, 1, 1, 4, 1, 6, 1, 13, 4, 6, 1, 25, 1, 6, 6, 38, 1, 26, 1, 26, 6, 6
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OFFSET
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1,4
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COMMENTS
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A positive subset-sum is a pair (h,g), where h is a positive integer and g is an integer partition, such that some submultiset of g sums to h. A triangle consists of a root sum r and a sequence of positive subset-sums ((h_1,g_1),...,(h_k,g_k)) such that the sequence (h_1,...,h_k) is weakly decreasing and has a submultiset summing to r. The composite of a triangle is (r, g_1 + ... + g_k) where + is multiset union.
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LINKS
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EXAMPLE
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We write positive subset-sum triangles in the form rootsum(branch,...,branch). The a(8) = 13 triangles:
1(1(1,1,1))
2(2(1,1,1))
3(3(1,1,1))
1(1(1),1(1,1))
2(1(1),1(1,1))
1(1(1),2(1,1))
2(1(1),2(1,1))
3(1(1),2(1,1))
1(1(1,1),1(1))
2(1(1,1),1(1))
1(1(1),1(1),1(1))
2(1(1),1(1),1(1))
3(1(1),1(1),1(1))
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CROSSREFS
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Cf. A063834, A262671, A269134, A276024, A281113, A299701, A301934, A301935, A316219, A316220, A316222.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A319001
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Number of ordered multiset partitions of integer partitions of n where the sequence of GCDs of the partitions is weakly increasing.
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+10
6
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1, 1, 3, 7, 18, 42, 105, 248, 606, 1450, 3507, 8415, 20305, 48785, 117502, 282574, 680137, 1636005, 3936841, 9470776, 22787529, 54822530, 131901491, 317336519, 763489051, 1836862947, 4419324581, 10632404189, 25580507505, 61543948594, 148068421107
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OFFSET
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0,3
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COMMENTS
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If we form a multiorder by treating integer partitions (a,...,z) as multiarrows GCD(a, ..., z) <= {z, ..., a}, then a(n) is the number of triangles of weight n.
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LINKS
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EXAMPLE
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The a(4) = 18 ordered multiset partitions:
{{4}} {{1,3}} {{2,2}} {{1,1,2}} {{1,1,1,1}}
{{1},{3}} {{2},{2}} {{1},{1,2}} {{1},{1,1,1}}
{{1,2},{1}} {{1,1,1},{1}}
{{1,1},{2}} {{1,1},{1,1}}
{{1},{1},{2}} {{1},{1},{1,1}}
{{1},{1,1},{1}}
{{1,1},{1},{1}}
{{1},{1},{1},{1}}
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PROG
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(PARI) \\ here B(n) is A000837 as vector.
B(n) = {dirmul(vector(n, k, moebius(k)), vector(n, k, numbpart(k)))}
seq(n) ={my(p=x*Ser(B(n))); Vec(1/prod(g=1, n, 1 - subst(p + O(x*x^(n\g)), x, x^g)))} \\ Andrew Howroyd, Jan 16 2023
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CROSSREFS
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Cf. A000837, A007716, A055887, A063834, A255397, A269134, A276024, A289508, A316222, A317545, A317546, A319002, A319003.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 16 2023
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STATUS
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approved
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A319003
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Number of ordered multiset partitions of integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.
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+10
4
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1, 1, 3, 7, 17, 38, 87, 191, 420, 908, 1954, 4160, 8816, 18549, 38851, 80965, 168077, 347566, 716443, 1472344, 3017866, 6170789, 12590805, 25640050, 52122784, 105791068, 214413852, 434007488, 877480395, 1772235212, 3575967030, 7209301989, 14523006820
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graph;
refs;
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history;
text;
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OFFSET
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0,3
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COMMENTS
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If we form a multiorder by treating integer partitions (a,...,z) as multiarrows LCM(a,...,z) <= {z,...,a}, then a(n) is the number of triangles of weight n.
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LINKS
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EXAMPLE
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The a(4) = 17 ordered multiset partitions:
{{4}} {{1,3}} {{2,2}} {{1,1,2}} {{1,1,1,1}}
{{1},{3}} {{2},{2}} {{1},{1,2}} {{1},{1,1,1}}
{{1,1},{2}} {{1,1,1},{1}}
{{1},{1},{2}} {{1,1},{1,1}}
{{1},{1},{1,1}}
{{1},{1,1},{1}}
{{1,1},{1},{1}}
{{1},{1},{1},{1}}
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PROG
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(PARI) seq(n)={my(M=Map()); for(m=1, n, forpart(p=m, my(k=lcm(Vec(p)), z); mapput(M, k, if(mapisdefined(M, k, &z), z, 1 + O(x*x^n)) - x^m))); Vec(1/vecprod(Mat(M)[, 2]))} \\ Andrew Howroyd, Jan 16 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 16 2023
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STATUS
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approved
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