A319193 - OEIS

%I #20 May 10 2021 04:01:47

%S 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 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 7,1,1,2,2,2,3,2,6,3,3,4,6,6,1,12,12,4,12

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

%C 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 Alois P. Heinz, <a href="/A319193/b319193.txt">Rows n = 0..33, flattened</a>

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

%e Triangle begins:

%e   1

%e   1

%e   1  1

%e   1  2  1

%e   1  1  2  3  1

%e   1  2  2  3  3  4  1

%e   1  2  2  1  1  3  6  6  4  5  1

%e The fourth row corresponds to the symmetric function identity: h(4) = -e(4) + e(22) + 2 e(31) - 3 e(211) + e(1111).

%p b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n], [seq(

%p       map(p-> p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])

%p     end:

%p T:= n-> map(m-> (l-> add(i, i=l)!/mul(i!, i=l))(map(

%p         i-> i[2], ifactors(m)[2])), sort(b(n$2)))[]:

%p seq(T(n), n=0..10);  # _Alois P. Heinz_, Feb 14 2020

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Length[Permutations[primeMS[k]]],{n,6},{k,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

%t (* Second program: *)

%t 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 T[n_] := Map[Function[m, Function[l, Total[l]!/Times @@ (l!)][ FactorInteger[m][[All, 2]]]], Sort[b[n, n]]];

%t T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)

%Y A different row ordering is A072811.

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

%K nonn,look,tabf

%O 0,6

%A _Gus Wiseman_, Sep 13 2018

%E T(0,1)=1 prepended by _Alois P. Heinz_, Feb 14 2020