A316223 -id:A316223 - cp's OEIS frontend

A316245 Number of ways to split an integer partition of n into consecutive subsequences with weakly decreasing sums.

1, 1, 3, 6, 14, 25, 52, 89, 167, 279, 486, 786, 1322, 2069, 3326, 5128, 8004, 12055, 18384, 27203, 40588, 59186, 86645, 124583, 179784, 255111, 362767, 509319, 715422, 993681, 1380793, 1899630, 2613064, 3564177, 4857631, 6572314, 8884973, 11930363, 16002853

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(4) = 14 split partitions:
  (4)
  (31)
  (22)
  (211)
  (3)(1)
  (2)(2)
  (1111)
  (21)(1)
  (2)(11)
  (111)(1)
  (11)(11)
  (2)(1)(1)
  (11)(1)(1)
  (1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(5) = 25 split partitions.
Gus Wiseman, The a(6) = 52 split partitions.

Cf. A000041, A001970, A063834, A316223, A317546, A319001, A319003.

Cf. A317508, 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[Sum[Length[Select[comps[y],OrderedQ[Total/@#,GreaterEqual]&]],{y,IntegerPartitions[n]}],{n,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f, self()(r,min(m,t),t,0,0)) + self()(r,m-1,s,t,0) + if(t+m<=s, self()(r-m,min(m,r-m),s,t+m,1)))); recurse(n,n,n,0,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

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

Original entry on oeis.org

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

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}]

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}]

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

cp's OEIS Frontend

A316245 Number of ways to split an integer partition of n into consecutive subsequences with weakly decreasing sums.

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

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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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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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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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