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
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
KEYWORD
sign
AUTHOR
George Beck, May 07 2017
STATUS
approved