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A305940
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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).
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4
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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
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OFFSET
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1
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COMMENTS
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Row n contains n nonzero entries, half of which (rounded up) are 1 and the remainder are -1.
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LINKS
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Table of n, a(n) for n=1..81.
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FORMULA
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T(n,k) = (-1)^(n - A061395(k)) if A215366(n,k) belongs to A093641 (Heinz numbers of hooks), 0 otherwise.
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EXAMPLE
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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)
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MATHEMATICA
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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]]}]
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CROSSREFS
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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
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KEYWORD
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sign,tabf
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AUTHOR
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Gus Wiseman, Sep 14 2018
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STATUS
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approved
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