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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
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.
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
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Sep 12 2018
STATUS
approved