A319003 Number of ordered multiset partitions of integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.

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

Offset: 0

Author

Gus Wiseman, Sep 07 2018

Keywords

• nonn

Comments

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.

Examples

			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}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A000837, A007716, A063834, A269134, A290103, A316222, A317545, A317546, A319001, A319004.

Programs

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

Extensions

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 16 2023