

A305940


Irregular triangle where T(n,k) is the coefficient of s(y) in p(n), where s is Schur functions, p is powersum 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
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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)^(nMax[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



