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

1,6

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.

Links

Table of n, a(n) for n=1..86.

Wikipedia, Symmetric polynomial

Example

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).

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}]

Crossrefs

Row sums are A321745.

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

Sequence in context: A302247 A026791 A080576 * A322763 A213211 A294775

Adjacent sequences: A321741 A321742 A321743 * A321745 A321746 A321747

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 19 2018

Status

approved