A286349 Möbius (or Moebius) partition function of partitions listed in the Abramowitz-Stegun order.

-1, -1, 0, -1, 1, 0, -1, 1, 0, 0, 0, -1, 1, 1, 0, 0, 0, 0, -1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 1, 0, -1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1

Comments

The sequence of row lengths of this array is [1,2,3,5,7,11,15,22,30,42,56,77,...] from A000041(n), n>=1 (partition numbers).

Links

Table of n, a(n) for n=1..96.

Ken Ono, Robert Schneider, and Ian Wagner, Partition-Theoretic Formulas for Arithmetic Densities, arXiv:1704.06636 [math.CO], April 21 2017.

Formula

a(n,k) = Möbius partition function of the k-th partition of n in Abramowitz-Stegun order (see reference). The Möbius partition function muP(p) of a partition p is defined by: muP(p) = (-1)^k if p has k distinct parts; otherwise muP(p) = 0 (p in the table of Abramowitz-Stegun).

Example

[-1];
[-1,0];
[-1,1,0];
[-1,1,0,0,0];
[-1,1,1,0,0,0,0];
[-1,1,1,0,0,-1,0,0,0,0,0];
...
Row 5 for partitions of 5 in the mentioned order: 5, 41, 32, 311, 221, 2111, 11111 with Möbius partition function values -1,1,1,0,0,0,0 because 5 has one part, 31 and 32 have two parts, and the rest have repeated parts.

Mathematica

PartitionMu[p_] := 0 /; Sort@p != Union@p;
PartitionMu[p_] := (-1)^Length@p /; Sort@p == Union@p;
table@T_ :=
Map[
PartitionMu,
  Table[
   Apply[Join, Reverse@*Sort /@ Table[IntegerPartitions[n, {k}],
   {k, n}]
  ],
  {n, T}],
{2}];
Flatten@table@10

Crossrefs

Row sums are A010815. - Seiichi Manyama, May 10 2017

Cf. A000009, A032020, A007837, A008683.

Sequence in context: A014555 A188017 A286755 * A145575 A077605 A323512

Adjacent sequences: A286346 A286347 A286348 * A286350 A286351 A286352

Keyword

sign

Author

George Beck, May 07 2017

Status

approved