A321744 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in h(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 6, 1, 3, 2, 4, 6, 1, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 7, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 1, 2, 3, 5, 4, 7, 10, 1, 6, 4, 12, 24, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Nov 19 2018

Keywords

• nonn
• 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).

Also the number of size-preserving permutations of type-v multiset partitions of a multiset whose multiplicities are the parts of u.

Also the coefficient of f(v) in e(u), where e is elementary symmetric functions and f is forgotten symmetric functions.

Examples

			Triangle begins:
   1
   1
   1   1
   1   2
   1   1   1
   1   2   3
   1   1   1   1   1
   1   3   6
   1   3   2   4   6
   1   2   2   3   4
   1   1   1   1   1   1   1
   1   4   3   7  12
   1   1   1   1   1   1   1   1   1   1   1
   1   2   2   3   3   4   5
   1   2   3   5   4   7  10
   1   6   4  12  24
   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
   1   3   5  11   8  18  30
For example, row 12 gives: h(211) = m(4) + 4m(22) + 3m(31) + 7m(211) + 12m(1111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are A321745.

Cf. A005651, A007716, A008480, A056239, A124794, A124795, A255906, A300121, A321742-A321765, A321854.

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,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn,Greater]]/Times@@Factorial/@Length/@Split[mtn],{mtn,Select[mps[nrmptn[n]],Sort[Length/@#]==primeMS[k]&]}],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}],{n,18}]