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).

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

Author

Gus Wiseman, Sep 12 2018

Keywords

• nonn
• tabf

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.

Examples

			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).
		

Links

• Wikipedia, Combinatorics of the Faà di Bruno coefficients

Crossrefs

Other row orderings are A036040, A080575, A178867.

Cf. A000041, A000110, A000258, A000670, A005651, A008277, A008480, A056239, A124794, A215366.

Programs

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}]

Formula

T(n,k) = A124794(A215366(n,k)).