A321899 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in F(u), where H is Heinz number, F is augmented forgotten 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

1,18

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 forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

Links

Table of n, a(n) for n=1..83.

Wikipedia, Symmetric polynomial

Example

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: F(211) = -2p(4) - p(22) - 2p(31) - p(211).

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[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*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]]]]}]

Crossrefs

Row sums are A130675, up to sign. Same as A321895, up to sign.

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

Sequence in context: A321900 A352195 A321895 * A027870 A357526 A070077

Adjacent sequences: A321896 A321897 A321898 * A321900 A321901 A321902

Keyword

sign,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved