A321895 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in M(u), where H is Heinz number, M is augmented monomial symmetric functions, and p is power sum symmetric functions.

1, 1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 2, -3, 1, -1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, -1, -2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, -6, 3, 8, -6, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Nov 20 2018

Keywords

• sign
• tabf

Comments

Row n has length A000041(A056239(n)).

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

The augmented monomial symmetric functions are given by M(y) = c(y) * m(y) where c(y) = Product_i (y)_i! where (y)_i is the number of i's in y and m is monomial symmetric functions.

Examples

			Triangle begins:
   1
   1
   1   0
  -1   1
   1   0   0
  -1   1   0
   1   0   0   0   0
   2  -3   1
  -1   1   0   0   0
  -1   0   1   0   0
   1   0   0   0   0   0   0
   2  -1  -2   1   0
   1   0   0   0   0   0   0   0   0   0   0
  -1   1   0   0   0   0   0
  -1   0   1   0   0   0   0
  -6   3   8  -6   1
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   2  -1  -2   1   0   0   0
For example, row 12 gives: M(211) = 2p(4) - p(22) - 2p(31) + p(211).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are A080339.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319193, A321742-A321765.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Product[(-1)^(Length[t]-1)*(Length[t]-1)!,{t,s}],{s,Select[sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},primeMS[n]][[i]],{i,PrimeOmega[n]}],Times@@Prime/@Total/@#==m&]}],{n,18},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]