A319001 Number of ordered multiset partitions of integer partitions of n where the sequence of GCDs of the partitions is weakly increasing.

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

Offset: 0

Author

Gus Wiseman, Sep 07 2018

Keywords

• nonn

Comments

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.

Examples

			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}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000837, A007716, A055887, A063834, A255397, A269134, A276024, A289508, A316222, A317545, A317546, A319002, A319003.

Programs

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

Extensions

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