A316222 -id:A316222 - cp's OEIS frontend

A316223 Number of subset-sum triangles with composite a subset-sum of the integer partition with Heinz number n.

0, 1, 1, 4, 1, 6, 1, 13, 4, 6, 1, 25, 1, 6, 6, 38, 1, 26, 1, 26, 6, 6

Offset: 1

Gus Wiseman, Jun 27 2018

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.

			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))

Cf. A063834, A262671, A269134, A276024, A281113, A299701, A301934, A301935, A316219, A316220, A316222.

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

Original entry on oeis.org

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

Gus Wiseman, Sep 07 2018

nonn

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.

			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}}

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

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

\\ 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

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

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

Original entry on oeis.org

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

Gus Wiseman, Sep 07 2018

nonn

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.

			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}}

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

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

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

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

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A316223 Number of subset-sum triangles with composite a subset-sum of the integer partition with Heinz number n.

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A319001 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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A319003 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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