A299202 Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.

0, 1, 1, -1, 1, -1, 1, 0, -1, -1, 1, 2, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 0, 1, 1, 3, 1, 0, -1, -1, -1, -1, 1, -1, -1, -1, 1, 2, 1, 1, 1, -1, 1, 0, -1, 1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -3, 1, -1, 2, 0, -1, 2, 1, 1, -1, 3, 1, 2, 1, -1, 1, 1, -1, 2, 1, 1, -1, -1, 1, -5, -1, -1, -1, -1, 1, -4

Offset

1,12

Comments

By convention, mu() = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=1..90.

Gus Wiseman, Comcategories and Multiorders

Formula

mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all enriched p-trees (A289501, A299203) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.

Example

Heinz number of (2,1,1) is 12, so mu(2,1,1) = a(12) = 2.

Mathematica

nn=120;
ptns=Table[If[n===1, {}, Join@@Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]], {n, nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&, ptns];
mu[y_]:=mu[y]=If[Length[y]===1, 1, -Sum[Times@@mu/@t, {t, Select[tris, And[Length[#]>1, Sort[Join@@#, Greater]===y]&]}]];
mu/@ptns

Crossrefs

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299203.

Sequence in context: A194325 A300547 A025452 * A194337 A365335 A299912

Adjacent sequences: A299199 A299200 A299201 * A299203 A299204 A299205

Keyword

sign

Author

Gus Wiseman, Feb 05 2018

Status

approved