A317545 -id:A317545 - cp's OEIS frontend

A317715 Number of ways to split an integer partition of n into consecutive subsequences with equal sums.

1, 1, 3, 4, 9, 8, 21, 16, 39, 38, 64, 57, 146, 102, 186, 211, 352, 298, 593, 491, 906, 880, 1273, 1256, 2444, 1998, 3038, 3277, 4861, 4566, 7710, 6843, 10841, 10742, 14966, 15071, 24499, 21638, 31334, 32706, 47157, 44584, 67464, 63262, 91351, 94247, 125248

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(4) = 9 constant-sum split partitions:
  (4),
  (31),
  (22), (2)(2),
  (211), (2)(11),
  (1111), (11)(11), (1)(1)(1)(1).
The a(6) = 21 constant-sum split partitions:
  (6),
  (51),
  (42),
  (411),
  (33), (3)(3),
  (321), (3)(21),
  (3111), (3)(111),
  (222), (2)(2)(2),
  (2211), (2)(2)(11),
  (21111), (21)(111), (2)(11)(11),
  (111111), (111)(111), (11)(11)(11), (1)(1)(1)(1)(1)(1).

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..500
Gus Wiseman, The a(8) = 39 constant-sum split partitions.
Gus Wiseman, The a(10) = 64 constant-sum split partitions.
Gus Wiseman, The a(12) = 146 constant-sum split partitions.

Cf. A000041, A001970, A063834, A316223, A317545, A317546, A319001.

Cf. A316245, A317508, A318434, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],SameQ@@Total/@#&]],{y,IntegerPartitions[n]}],{n,10}]

a(16)-a(46) from Hiroaki Yamanouchi, Oct 02 2018

A317546 Number of multimin partitions of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 18, 42, 104, 246, 594, 1416, 3391, 8084, 19312, 46041, 109829, 261827, 624254, 1487981, 3546883, 8453770, 20149014, 48021864, 114451536, 272769936, 650084053, 1549312743

Offset: 1

Gus Wiseman, Jul 31 2018

nonn

A multimin partition of m is an ordered multiset partition of m such that the minima of the blocks are weakly increasing.

			The a(3) = 7 multimin partitions of integer partitions of 3:
  (3),
  (1)(2), (12),
  (1)(1)(1), (1)(11), (11)(1), (111).
The a(4) = 18 multimin partitions of integer partitions of 4:
  (4),
  (1)(3), (13),
  (2)(2), (22),
  (1)(1)(2), (1)(12), (11)(2), (12)(1), (112),
  (1)(1)(1)(1), (1)(1)(11), (1)(11)(1), (1)(111), (11)(1)(1), (11)(11), (111)(1), (1111).

Cf. A007716, A020639, A034691, A255397, A300335, A317545.

mmcount[m_List]:=mmcount[m]=If[Length[m]===0,0,1+Plus@@mmcount/@Union[Subsets[Rest[m]]]];
Table[Sum[mmcount[Reverse[ptn]],{ptn,IntegerPartitions[n]}],{n,25}]

a(n) = Sum_{k > 0 : A056239(k) = n} A317545(k).

A318434 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(3072) = 5 constant-sum split partitions:
  (21111111111)
  (21111)(111111)
  (211)(1111)(1111)
  (21)(111)(111)(111)
  (2)(11)(11)(11)(11)(11)

Cf. A001970, A056239, A063834, A296150, A316223, A317545, A317546, A319002.

Cf. A316245, A317508, A317715, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Length[Select[comps[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],SameQ@@Total/@#&]],{n,100}]

A317508 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with weakly decreasing sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 4, 1, 7, 2, 2, 2, 6, 1, 2, 2, 7, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 5, 2, 7, 2, 2, 1, 7, 1, 2, 4, 11, 2, 4, 1, 4, 2, 4, 1, 9, 1, 2, 3, 4, 2, 4, 1, 11, 5, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(60) = 7 split partitions:
  (3)(2)(1)(1)
  (32)(1)(1)
  (3)(21)(1)
  (3)(2)(11)
  (321)(1)
  (32)(11)
  (3211)

Cf. A001970, A056239, A063834, A255397, A296150, A316223, A317545, A317546, A319002, A319004.

Cf. A316245, A317715, A318434, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Length[Select[compositionPartitions[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],OrderedQ[Total/@#]&]],{n,100}]

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

A319002 Number of ordered factorizations of n where the sequence of GCDs of prime indices (A289508) of the factors is weakly increasing.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 4, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 6, 1, 16, 2, 2, 2, 11, 1, 2, 2, 12, 1, 5, 1, 5, 4, 2, 1, 28, 2, 4, 2, 5, 1, 8, 2, 12, 2, 2, 1, 18, 1, 2, 5, 32, 2, 6, 1, 5, 2, 6, 1, 29, 1, 2, 4, 5, 2, 5, 1, 28, 8, 2, 1

Offset: 1

Gus Wiseman, Sep 07 2018

nonn

Also the number of ordered multiset partitions of the multiset of prime indices of n where the sequence of GCDs of the blocks is weakly increasing. 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 whose composite ground is the integer partition with Heinz number n.

			The a(36) = 11 ordered factorizations:
  (2*2*3*3),
  (2*2*9), (2*6*3), (6*2*3), (4*3*3),
  (2*18), (18*2), (12*3), (4*9), (6*6),
  (36).
The a(36) = 11 ordered multiset partitions:
     {{1,1,2,2}}
    {{1},{1,2,2}}
    {{1,2,2},{1}}
    {{1,1,2},{2}}
    {{1,1},{2,2}}
    {{1,2},{1,2}}
   {{1},{1},{2,2}}
   {{1},{1,2},{2}}
   {{1,2},{1},{2}}
   {{1,1},{2},{2}}
  {{1},{1},{2},{2}}

Cf. A000837, A007716, A001970, A056239, A063834, A289508, A296150, A316223, A317545, A318559, A319001, A319004.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
gix[n_]:=GCD@@PrimePi/@If[n==1,{},FactorInteger[n]][[All,1]];
Table[Length[Select[Join@@Permutations/@facs[n],OrderedQ[gix/@#]&]],{n,100}]

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

A319004 Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 8, 1, 5, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 16, 2, 2, 2, 11, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 16, 2, 5, 2, 4, 1, 12, 2, 8, 2, 2, 1, 11, 1, 2, 4, 32, 2, 5, 1, 4, 2, 5, 1, 23, 1, 2, 4, 4, 2, 5, 1, 16, 8, 2, 1, 11, 2, 2, 2, 8, 1, 12, 2, 4, 2, 2, 2, 32, 1, 5, 4, 11, 1, 5, 1, 8, 5

Offset: 1

Gus Wiseman, Sep 07 2018

nonn

Also the number of ordered multiset partitions of the multiset of prime indices of n where the sequence of LCMs of the parts is weakly increasing. 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 whose composite ground is the integer partition with Heinz number n.

			The a(60) = 11 ordered factorizations:
  (2*2*3*5),
  (2*2*15), (2*3*10), (2*6*5), (4*3*5),
  (2*30), (3*20), (4*15), (12*5), (6*10),
  (60).
The a(60) = 11 ordered multiset partitions:
     {{1,1,2,3}}
    {{1},{1,2,3}}
    {{2},{1,1,3}}
    {{1,1,2},{3}}
    {{1,1},{2,3}}
    {{1,2},{1,3}}
   {{1},{1},{2,3}}
   {{1},{2},{1,3}}
   {{1},{1,2},{3}}
   {{1,1},{2},{3}}
  {{1},{1},{2},{3}}

Cf. A001055, A056239, A063834, A074206, A290103, A296150, A316223, A317545, A317546, A318559, A319002, A319003.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
lix[n_]:=LCM@@PrimePi/@If[n==1,{},FactorInteger[n]][[All,1]];
Table[Length[Select[Join@@Permutations/@facs[n],OrderedQ[lix/@#]&]],{n,100}]

is_weakly_increasing(v) = { for(i=2,#v,if(v[i]A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A319004aux(n, facs) = if(1==n, is_weakly_increasing(apply(f -> A290103(f),Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1), newfacs = List(facs); listput(newfacs,d); s += A319004aux(n/d, newfacs))); (s));
A319004(n) = if((1==n)||isprime(n),1,A319004aux(n, List([]))); \\ Antti Karttunen, Sep 23 2018

A001055(n) <= a(n) <= A074206(n). - Antti Karttunen, Sep 23 2018

More terms from Antti Karttunen, Sep 23 2018

A319118 Number of multimin tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 8, 1, 2, 2, 24, 1, 6, 1, 8, 2, 2, 1, 42, 2, 2, 6, 8, 1, 8, 1, 112, 2, 2, 2, 38, 1, 2, 2, 42, 1, 8, 1, 8, 8, 2, 1, 244, 2, 6, 2, 8, 1, 24, 2, 42, 2, 2, 1, 58, 1, 2, 8, 568, 2, 8, 1, 8, 2, 8, 1, 268, 1, 2, 6, 8, 2, 8, 1, 244, 24

Offset: 1

Gus Wiseman, Sep 10 2018

nonn

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of multimin tree-factorizations, one of each factor in a multimin factorization of n with at least two factors.

			The a(12) = 8 multimin tree-factorizations:
  12,
  (2*6), (4*3), (6*2), (2*2*3),
  (2*(2*3)), ((2*2)*3), ((2*3)*2).
Or as series-reduced plane trees of multisets:
  112,
  (1,12), (11,2), (12,1), (1,1,2),
  (1,(1,2)), ((1,1),2), ((1,2),1).

Cf. A001055, A005804, A020639, A118376, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A318577, A319119.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Length[mmftrees[n]],{n,100}]

a(prime^n) = A118376(n).

a(product of n distinct primes) = A005804(n).

A318577 Number of complete multimin tree-factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 11, 1, 3, 1, 4, 1, 1, 1, 19, 1, 1, 3, 4, 1, 4, 1, 45, 1, 1, 1, 17, 1, 1, 1, 19, 1, 4, 1, 4, 4, 1, 1, 96, 1, 3, 1, 4, 1, 11, 1, 19, 1, 1, 1, 26, 1, 1, 4, 197, 1, 4, 1, 4, 1, 4, 1, 104, 1, 1, 3, 4, 1, 4, 1, 96, 11, 1, 1, 26, 1, 1, 1, 19, 1, 19, 1, 4, 1, 1, 1, 501, 1, 3, 4, 17

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of at least two multimin tree-factorizations, one of each factor in a multimin factorization of n. A multimin tree-factorization is complete if the leaves are all prime numbers.

			The a(12) = 4 trees are (2*2*3), (2*(2*3)), ((2*3)*2), ((2*2)*3).

Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A319118, A319121.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Length[Select[mmftrees[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A001003(n - 1).

a(product of n distinct primes) = A000311(n).

cp's OEIS Frontend

A317715 Number of ways to split an integer partition of n into consecutive subsequences with equal sums.

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A317546 Number of multimin partitions of integer partitions of n.

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A318434 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with equal sums.

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A317508 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with weakly decreasing sums.

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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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A319002 Number of ordered factorizations of n where the sequence of GCDs of prime indices (A289508) of the factors 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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A319004 Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.

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