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A319193
Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).
45
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
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.
Sequence in context: A241898 A191090 A331958 * A097886 A308293 A249298
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