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A319193 Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k). 44
1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 1, 3, 6, 6, 4, 5, 1, 1, 2, 2, 2, 6, 3, 3, 3, 4, 4, 12, 10, 5, 6, 1, 1, 2, 2, 1, 3, 2, 3, 6, 6, 3, 1, 12, 4, 12, 6, 10, 5, 20, 15, 6, 7, 1, 1, 2, 2, 2, 3, 2, 6, 3, 3, 4, 6, 6, 1, 12, 12, 4, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A refinement of Pascal's triangle, these are the unsigned coefficients appearing in the expansion of homogeneous symmetric functions in terms of elementary symmetric functions.

LINKS

Table of n, a(n) for n=1..83.

FORMULA

a(n,k) = A008480(A215366(n,k)).

EXAMPLE

Triangle begins:

  1

  1  1

  1  2  1

  1  1  2  3  1

  1  2  2  3  3  4  1

  1  2  2  1  1  3  6  6  4  5  1

The fourth row corresponds to the symmetric function identity: h(4) = -e(4) + e(22) + 2 e(31) - 3 e(211) + e(1111).

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Length[Permutations[primeMS[k]]], {n, 6}, {k, Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

CROSSREFS

A different row ordering is A072811.

Cf. A000041, A005651, A008277, A008480, A056239, A124794, A215366, A319182, A319191, A319192.

Sequence in context: A301314 A241898 A191090 * A097886 A308293 A249298

Adjacent sequences:  A319190 A319191 A319192 * A319194 A319195 A319196

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Sep 13 2018

STATUS

approved

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Last modified November 14 00:05 EST 2019. Contains 329106 sequences. (Running on oeis4.)