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%I #7 Sep 15 2018 15:48:34
%S 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 1,1,7,21,35,105,21,105,70,105,35,210,105,35,21,1,1,8,28,35,28,56,210,
%U 168,280,280,105,420,56,840,280,420,70,560,210,56,28,1,1
%N 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 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 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 T(n,k) = A124794(A215366(n,k)).
%e Triangle begins:
%e 1
%e 1 1
%e 1 3 1
%e 1 3 4 6 1
%e 1 5 10 15 10 10 1
%e 1 15 6 10 15 15 60 45 20 15 1
%e 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 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
%t Table[numSetPtnsOfType/@primeMS/@Sort[Times@@Prime/@#&/@IntegerPartitions[n]],{n,7}]
%Y Other row orderings are A036040, A080575, A178867.
%Y Cf. A000041, A000110, A000258, A000670, A005651, A008277, A008480, A056239, A124794, A215366.
%K nonn,tabf
%O 1,5
%A _Gus Wiseman_, Sep 12 2018
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