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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

Table of n, a(n) for n=1..81.

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

Cf. A056239, A082733, A093641, A153452, A153734, A215366, A296188, A296561.

Cf. A304438, A305940, A317552, A317553, A317554.

Sequence in context: A118172 A071039 A118111 * A119981 A115789 A053864

Adjacent sequences: A305937 A305938 A305939 * A305941 A305942 A305943

KEYWORD

sign,tabf

AUTHOR

Gus Wiseman, Sep 14 2018

STATUS

approved

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Last modified November 26 17:13 EST 2022. Contains 358362 sequences. (Running on oeis4.)