|
%I #18 Dec 26 2018 16:56:21
%S -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,
%T 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,
%U 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
%N Möbius (or Moebius) partition function of partitions listed in the Abramowitz-Stegun order.
%C 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).
%H Ken Ono, Robert Schneider, and Ian Wagner, <a href="https://arxiv.org/abs/1704.06636">Partition-Theoretic Formulas for Arithmetic Densities</a>, arXiv:1704.06636 [math.CO], April 21 2017.
%F 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).
%e [-1];
%e [-1,0];
%e [-1,1,0];
%e [-1,1,0,0,0];
%e [-1,1,1,0,0,0,0];
%e [-1,1,1,0,0,-1,0,0,0,0,0];
%e ...
%e 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.
%t PartitionMu[p_] := 0 /; Sort@p != Union@p;
%t PartitionMu[p_] := (-1)^Length@p /; Sort@p == Union@p;
%t table@T_ :=
%t Map[
%t PartitionMu,
%t Table[
%t Apply[Join, Reverse@*Sort /@ Table[IntegerPartitions[n, {k}],
%t {k, n}]
%t ],
%t {n, T}],
%t {2}];
%t Flatten@table@10
%Y Row sums are A010815. - _Seiichi Manyama_, May 10 2017
%Y Cf. A000009, A032020, A007837, A008683.
%K sign
%O 1
%A _George Beck_, May 07 2017
|
