A318434 -id:A318434 - 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

A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Apr 07 2024

nonn

These partitions are counted by A371795, even case A006827.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 aerated and ranked by A357976.

			The prime indices of 975 are {2,3,3,6}, which are not biquanimous, so 975 is in the sequence.
The prime indices of 900 are {1,1,2,2,3,3}, which can be partitioned into {{1,2,3},{1,2,3}} or {{3,3},{1,1,2,2}}, so 900 is not in the sequence.

The complement is A357976, counted by A002219.
For prime signature instead of indices we have A371782, complement A371781.
Partitions of this type are counted by A371795, even case A006827.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A000005, A002221, A027187, A035470, A064914, A299701, A300061, A318434, A321455, A326534, A357879, A366320, A367094.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Select[Range[100],Not@*biqQ@*prix]

Numbers n without a divisor d|n such that A056239(d) = A056239(n/d).

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

A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

Original entry on oeis.org

0, 1, 3, 6, 12, 24, 46, 90, 174, 337, 651, 1261, 2445, 4753, 9258, 18101, 35487, 69823, 137704, 272366, 539797, 1071969, 2132017, 4245964, 8464289, 16887427, 33713589, 67336900, 134542546, 268894341, 537515903, 1074640717, 2148733325, 4296686409, 8592299548, 17183084263, 34364120060, 68725368752, 137446915007, 274888501928, 549770021804, 1099530342380, 2199048203425, 4398079052052, 8796136153039, 17592241805077, 35184445671235

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The subsets of S = {1,4,6,7} have distinct sums {0,1,4,5,6,7,8,10,11,12,13,14,17,18}. Since 9 is missing, S is counted under a(7).
The a(0) = 0 through a(4) = 12 subsets:
  .  {1}  {1}    {1}    {1}
          {2}    {2}    {2}
          {1,2}  {3}    {3}
                 {1,2}  {4}
                 {1,3}  {1,2}
                 {2,3}  {1,3}
                        {1,4}
                        {2,3}
                        {2,4}
                        {3,4}
                        {1,2,4}
                        {2,3,4}

This is the "bi-" version of A371789, differences A371790.
The complement is counted by A371791, differences A232466.
First differences are A371793.
The complement is the "bi-" version of A371796, differences A371797.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A318434, A321455, A365543, A365661, A365663, A366320, A365381, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],Not@*biqQ]],{n,0,10}]

a(16) onwards from Martin Fuller, Mar 21 2025

A371794 Number of non-biquanimous strict integer partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 27, 23, 38, 30, 54, 43, 76, 57, 104, 79, 142, 102, 192, 138, 256, 174, 340, 232, 448, 292, 585, 375, 760, 471, 982, 602, 1260, 741, 1610, 935, 2048, 1148, 2590, 1425, 3264, 1733, 4097, 2137, 5120, 2571, 6378

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The a(1) = 1 through a(11) = 12 strict partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (521)  (81)   (91)   (92)
                                                  (432)  (631)  (A1)
                                                  (531)  (721)  (542)
                                                  (621)         (632)
                                                                (641)
                                                                (731)
                                                                (821)
                                                                (5321)

The complement is counted by A237258 aerated, ranks A357854.
Even bisection is A321142, odd A078408.
This is the "bi-" version of A371736, complement A371737.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A064914, A279787, A305551, A318434, A365543, A365663, A365661, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!biqQ[#]&]],{n,0,30}]

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}]

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}]

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A371731 Heinz numbers of non-biquanimous integer partitions. Numbers without a divisor having the same sum of prime indices as the quotient.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A371794 Number of non-biquanimous strict integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments