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A319182 Irregular triangle where T(n,k) is the number of set partitions of {1,...,n} with block-sizes given by the integer partition with Heinz number A215366(n,k).

%I A319182 #7 Sep 15 2018 15:48:34
%S A319182 1,1,1,1,3,1,1,3,4,6,1,1,5,10,15,10,10,1,1,15,6,10,15,15,60,45,20,15,
%T A319182 1,1,7,21,35,105,21,105,70,105,35,210,105,35,21,1,1,8,28,35,28,56,210,
%U A319182 168,280,280,105,420,56,840,280,420,70,560,210,56,28,1,1
%N A319182 Irregular triangle where T(n,k) is the number of set partitions of {1,...,n} with block-sizes given by the integer partition with Heinz number A215366(n,k).
%C A319182 A generalization of the triangle of Stirling numbers of the second kind, these are the coefficients appearing in the expansion of (x_1 + x_2 + x_3 + ...)^n in terms of augmented monomial symmetric functions. They also appear in Faa di Bruno's formula.
%H A319182 Wikipedia, <a href="https://en.wikipedia.org/wiki/Faà_di_Bruno%27s_formula#Combinatorics_of_the_Faà_di_Bruno_coefficients">Combinatorics of the Faà di Bruno coefficients</a>
%F A319182 T(n,k) = A124794(A215366(n,k)).
%e A319182 Triangle begins:
%e A319182   1
%e A319182   1   1
%e A319182   1   3   1
%e A319182   1   3   4   6   1
%e A319182   1   5  10  15  10  10   1
%e A319182   1  15   6  10  15  15  60  45  20  15   1
%e A319182 The fourth row corresponds to the symmetric function identity (x_1 + x_2 + x_3 + ...)^4 = m(4) + 3 m(22) + 4 m(31) + 6 m(211) + m(1111).
%t A319182 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A319182 numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
%t A319182 Table[numSetPtnsOfType/@primeMS/@Sort[Times@@Prime/@#&/@IntegerPartitions[n]],{n,7}]
%Y A319182 Other row orderings are A036040, A080575, A178867.
%Y A319182 Cf. A000041, A000110, A000258, A000670, A005651, A008277, A008480, A056239, A124794, A215366.
%K A319182 nonn,tabf
%O A319182 1,5
%A A319182 _Gus Wiseman_, Sep 12 2018