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

1,5

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

Table of n, a(n) for n=1..67.

Wikipedia, Combinatorics of the Faà di Bruno coefficients

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