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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). 10
 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, 1, 1, 7, 21, 35, 105, 21, 105, 70, 105, 35, 210, 105, 35, 21, 1, 1, 8, 28, 35, 28, 56, 210, 168, 280, 280, 105, 420, 56, 840, 280, 420, 70, 560, 210, 56, 28, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. LINKS FORMULA T(n,k) = A124794(A215366(n,k)). EXAMPLE Triangle begins:   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   1 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). MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn]; Table[numSetPtnsOfType/@primeMS/@Sort[Times@@Prime/@#&/@IntegerPartitions[n]], {n, 7}] CROSSREFS Other row orderings are A036040, A080575, A178867. Cf. A000041, A000110, A000258, A000670, A005651, A008277, A008480, A056239, A124794, A215366. Sequence in context: A097560 A218905 A027960 * A247282 A246685 A218618 Adjacent sequences:  A319179 A319180 A319181 * A319183 A319184 A319185 KEYWORD nonn,tabf AUTHOR Gus Wiseman, Sep 12 2018 STATUS approved

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Last modified May 26 22:58 EDT 2020. Contains 334634 sequences. (Running on oeis4.)