A317546 -id:A317546 - 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

A318684 Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 11, 14, 20, 28, 35, 48, 61, 79, 105, 129, 162, 208, 257, 318, 404, 489, 600, 732, 896, 1075, 1315, 1576, 1895, 2272, 2715, 3217, 3851, 4537, 5377, 6353, 7484, 8765, 10314, 12044, 14079, 16420, 19114, 22184, 25818, 29840, 34528, 39903, 46030

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(9) = 20 split partitions:
    (9)
   (81)   (8)(1)
   (72)   (7)(2)
   (63)   (6)(3)
   (54)   (5)(4)
  (432)  (43)(2)  (4)(3)(2)
  (621)  (62)(1)  (6)(2)(1)  (6)(21)
  (531)  (53)(1)  (5)(3)(1)  (5)(31)

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, 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/@#,Greater]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A319794 Number of ways to split a strict integer partition of n into consecutive subsequences with weakly decreasing sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 20, 31, 37, 52, 64, 85, 111, 141, 175, 225, 279, 346, 437, 532, 654, 802, 979, 1182, 1438, 1740, 2083, 2502, 2996, 3565, 4245, 5043, 5950, 7068, 8303, 9772, 11449, 13452, 15681, 18355, 21338, 24855, 28846, 33509, 38687, 44819, 51644

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(6) = 9 split partitions:
    (6)
   (51)  (5)(1)
   (42)  (4)(2)
  (321)  (32)(1)  (3)(21)  (3)(2)(1).

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A318684.

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,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A318683 Number of ways to split a strict integer partition of n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 5, 7, 8, 12, 12, 18, 18, 26, 27, 37, 38, 53, 54, 73, 76, 100, 104, 136, 142, 183, 192, 244, 256, 327, 340, 424, 448, 558, 585, 722, 760, 937, 983, 1195, 1260, 1544, 1610, 1943, 2053, 2480, 2590, 3107, 3264, 3927, 4106, 4874, 5120, 6134, 6378

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(12) = 18 constant-sum split partitions:
  (12)
  (7,5)
  (8,4)
  (9,3)
  (10,2)
  (11,1)
  (5,4,3)
  (6,4,2)
  (6,5,1)
  (7,3,2)
  (7,4,1)
  (8,3,1)
  (9,2,1)
  (6)(4,2)
  (6)(5,1)
  (5,4,2,1)
  (6,3,2,1)
  (6)(3,2,1)

Cf. A001970, A063834, A316245, A317508, A317546, 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],SameQ@@Total/@#&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A317545 Number of multimin factorizations of n.

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, 5, 1, 16, 2, 2, 2, 11, 1, 2, 2, 12, 1, 5, 1, 5, 5, 2, 1, 28, 2, 4, 2, 5, 1, 8, 2, 12, 2, 2, 1, 15, 1, 2, 5, 32, 2, 5, 1, 5, 2, 5, 1, 29, 1, 2, 4, 5, 2, 5, 1, 28, 8, 2, 1, 15, 2, 2, 2, 12, 1, 12, 2, 5, 2, 2, 2, 64, 1, 4, 5, 11, 1, 5, 1, 12, 5

Offset: 1

Gus Wiseman, Jul 31 2018

nonn

A multimin factorizations 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.

			The a(36) = 11 multimin factorizations:
  (36),
  (2*18), (4*9), (6*6), (12*3), (18*2),
  (2*2*9), (2*6*3), (4*3*3), (6*2*3),
  (2*2*3*3).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007716, A020639, A034691, A255397, A296560, A300335, A317546.

a[n_]:=If[n==1,1,Sum[a[d],{d,Divisors[n/FactorInteger[n][[1,1]]]}]];
Array[a,100]

A317545(n) = if(1==n,1,my(spf = factor(n)[1,1]); sumdiv(n/spf,d,A317545(d))); \\ Antti Karttunen, Sep 10 2018

memo317545 = Map(); \\ Memoized version.
A317545(n) = if(1==n,1,if(mapisdefined(memo317545, n), mapget(memo317545, n), my(spf = factor(n)[1,1], v = sumdiv(n/spf,d,A317545(d))); mapput(memo317545, n, v); (v))); \\ Antti Karttunen, Sep 10 2018

a(1) = 1; a(n > 1) = Sum_{d|(n/p)} a(d), where p is the smallest prime dividing n.

More terms from Antti Karttunen, Sep 10 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}]

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

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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A318684 Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.

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A319794 Number of ways to split a strict integer partition of n into consecutive subsequences with weakly decreasing sums.

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A318683 Number of ways to split a strict integer partition of n into consecutive subsequences with equal sums.

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Mathematica

A317545 Number of multimin factorizations 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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Mathematica

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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