A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).

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, 1, 2, 2, 2, 6, 3, 3, 3, 4, 4, 12, 10, 5, 6, 1, 1, 2, 2, 1, 3, 2, 3, 6, 6, 3, 1, 12, 4, 12, 6, 10, 5, 20, 15, 6, 7, 1, 1, 2, 2, 2, 3, 2, 6, 3, 3, 4, 6, 6, 1, 12, 12, 4, 12

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

Formula

T(n,k) = A008480(A215366(n,k)).

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

A different row ordering is A072811.

Cf. A000041, A005651, A008277, A008480, A056239, A124794, A215366, A319182, A319191, A319192.

Sequence in context: A241898 A191090 A331958 * A097886 A308293 A249298

Adjacent sequences: A319190 A319191 A319192 * A319194 A319195 A319196

Keyword

nonn,look,tabf

Author

Gus Wiseman, Sep 13 2018

Extensions

T(0,1)=1 prepended by Alois P. Heinz, Feb 14 2020

Status

approved