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A305940 Irregular triangle where T(n,k) is the coefficient of s(y) in p(n), where s is Schur functions, p is power-sum symmetric functions, and y is the integer partition with Heinz number A215366(n,k). 4
1, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, -1, 1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 1, -1, 1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
Row n contains n nonzero entries, half of which (rounded up) are 1 and the remainder are -1.
LINKS
FORMULA
T(n,k) = (-1)^(n - A061395(k)) if A215366(n,k) belongs to A093641 (Heinz numbers of hooks), 0 otherwise.
EXAMPLE
Triangle begins:
1
1 -1
1 -1 1
1 0 -1 1 -1
1 -1 0 0 1 -1 1
1 0 -1 0 0 1 0 0 -1 1 -1
1 -1 0 0 0 1 0 0 0 -1 0 0 1 -1 1
1 -1 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 -1 1 -1
The rows correspond to the following symmetric function identities.
p(1) = s(1)
p(2) = s(2) - s(11)
p(3) = s(3) - s(21) + s(111)
p(4) = s(4) - s(31) + s(211) - s(1111)
p(5) = s(5) - s(41) + s(311) - s(2111) + s(11111)
MATHEMATICA
hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n], {2, _}], {}|{{_, 1}}];
Table[If[hookQ[k], (-1)^(n-Max[PrimePi/@FactorInteger[k][[All, 1]]]), 0], {n, 8}, {k, Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]
CROSSREFS
Sequence in context: A118172 A071039 A118111 * A119981 A115789 A359471
KEYWORD
sign,tabf
AUTHOR
Gus Wiseman, Sep 14 2018
STATUS
approved

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)