A357847 -id:A357847 - cp's OEIS frontend

A357709 Number of integer partitions of n whose length is twice their alternating sum.

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 6, 6, 9, 11, 13, 18, 21, 28, 32, 44, 49, 65, 76, 96, 114, 141, 170, 204, 250, 295, 361, 425, 516, 606, 734, 858, 1031, 1210, 1440, 1690, 2000, 2347, 2759, 3240, 3786, 4441, 5174, 6053, 7030, 8210, 9509, 11074, 12807, 14870

Offset: 0

Gus Wiseman, Oct 16 2022

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The alternating sum of a partition is also the number of odd conjugate parts.

			The a(1) = 0 through a(12) = 6 partitions:
  .  .  21  .  32  3111  43  3221  54      3331  65      4332
                             4211  411111  4222  422111  4431
                                           4321  521111  5322
                                           5311          5421
                                                         6411
                                                         51111111

This is the "twice" version of A357189, ranked by A357486.
The version for compositions is A357847.
These partitions are ranked by A357848.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
A357182 counts compositions w/ length = alternating sum, ranked by A357184.
Cf. A004526, A262046, A262977, A301987, A357183, A357485, A357488.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==2ats[#]&]],{n,0,30}]

A357848 Heinz numbers of integer partitions whose length is twice their alternating sum.

Original entry on oeis.org

1, 6, 15, 35, 40, 77, 84, 90, 143, 189, 210, 220, 221, 224, 250, 323, 364, 437, 462, 490, 495, 504, 525, 528, 667, 748, 819, 858, 899, 988, 1029, 1040, 1134, 1147, 1155, 1188, 1210, 1320, 1326, 1375, 1400, 1408, 1517, 1564, 1683, 1690, 1763, 1904, 1938, 2021

Offset: 1

Gus Wiseman, Oct 16 2022

nonn

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.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms together with their prime indices begin:
     1: {}
     6: {1,2}
    15: {2,3}
    35: {3,4}
    40: {1,1,1,3}
    77: {4,5}
    84: {1,1,2,4}
    90: {1,2,2,3}
   143: {5,6}
   189: {2,2,2,4}
   210: {1,2,3,4}
   220: {1,1,3,5}
   221: {6,7}
   224: {1,1,1,1,1,4}

These partitions are counted by A357709.
The version for compositions is counted by A357847.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A025047 counts alternating compositions.
A056239 adds up prime indices.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
A357182 counts compositions w/ length = alternating sum, ranked by A357184.
A357189 counts partitions w/ length = alternating sum, ranked by A357486.
Cf. A000720, A001221, A001222, A262977, A301987, A357183, A357485, A357488.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[1000],Length[primeMS[#]]==2sats[primeMS[#]]&]

cp's OEIS Frontend

A357709 Number of integer partitions of n whose length is twice their alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica