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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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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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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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