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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).

%I A319193 #20 May 10 2021 04:01:47
%S A319193 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,
%T A319193 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,
%U A319193 7,1,1,2,2,2,3,2,6,3,3,4,6,6,1,12,12,4,12
%N A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).
%C A319193 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.
%H A319193 Alois P. Heinz, <a href="/A319193/b319193.txt">Rows n = 0..33, flattened</a>
%F A319193 T(n,k) = A008480(A215366(n,k)).
%e A319193 Triangle begins:
%e A319193   1
%e A319193   1
%e A319193   1  1
%e A319193   1  2  1
%e A319193   1  1  2  3  1
%e A319193   1  2  2  3  3  4  1
%e A319193   1  2  2  1  1  3  6  6  4  5  1
%e A319193 The fourth row corresponds to the symmetric function identity: h(4) = -e(4) + e(22) + 2 e(31) - 3 e(211) + e(1111).
%p A319193 b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n], [seq(
%p A319193       map(p-> p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
%p A319193     end:
%p A319193 T:= n-> map(m-> (l-> add(i, i=l)!/mul(i!, i=l))(map(
%p A319193         i-> i[2], ifactors(m)[2])), sort(b(n$2)))[]:
%p A319193 seq(T(n), n=0..10);  # _Alois P. Heinz_, Feb 14 2020
%t A319193 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A319193 Table[Length[Permutations[primeMS[k]]],{n,6},{k,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]
%t A319193 (* Second program: *)
%t A319193 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 A319193 T[n_] := Map[Function[m, Function[l, Total[l]!/Times @@ (l!)][ FactorInteger[m][[All, 2]]]], Sort[b[n, n]]];
%t A319193 T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y A319193 A different row ordering is A072811.
%Y A319193 Cf. A000041, A005651, A008277, A008480, A056239, A124794, A215366, A319182, A319191, A319192.
%K A319193 nonn,look,tabf
%O A319193 0,6
%A A319193 _Gus Wiseman_, Sep 13 2018
%E A319193 T(0,1)=1 prepended by _Alois P. Heinz_, Feb 14 2020