OFFSET
0,6
COMMENTS
A refinement of Pascal's triangle, these are the unsigned coefficients appearing in the expansion of homogeneous symmetric functions in terms of elementary symmetric functions.
LINKS
Alois P. Heinz, Rows n = 0..33, flattened
EXAMPLE
Triangle begins:
1
1
1 1
1 2 1
1 1 2 3 1
1 2 2 3 3 4 1
1 2 2 1 1 3 6 6 4 5 1
The fourth row corresponds to the symmetric function identity: h(4) = -e(4) + e(22) + 2 e(31) - 3 e(211) + e(1111).
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n], [seq(
map(p-> p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
end:
T:= n-> map(m-> (l-> add(i, i=l)!/mul(i!, i=l))(map(
i-> i[2], ifactors(m)[2])), sort(b(n$2)))[]:
seq(T(n), n=0..10); # Alois P. Heinz, Feb 14 2020
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Permutations[primeMS[k]]], {n, 6}, {k, Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0 || i < 2, {2^n}, Flatten[Table[ #*Prime[i]^j& /@ b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Map[Function[m, Function[l, Total[l]!/Times @@ (l!)][ FactorInteger[m][[All, 2]]]], Sort[b[n, n]]];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Gus Wiseman, Sep 13 2018
EXTENSIONS
T(0,1)=1 prepended by Alois P. Heinz, Feb 14 2020
STATUS
approved