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A296624
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a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(mu) with partition lambda >= mu and size(lambda) + size(mu)= n.
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3
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1, 1, 4, 7, 20, 37, 90, 171, 378, 721, 1500, 2843, 5682, 10661, 20674
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OFFSET
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0,3
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COMMENTS
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The condition lambda >= mu restricts the results to the lower triangular part of the matrix formed by products of all pairs of partitions.
'Multiplicity' signifies that terms like k*s(nu) count as k terms.
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LINKS
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Table of n, a(n) for n=0..14.
Wouter Meeussen, Mathematica toolbox for symmetric functions
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EXAMPLE
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For n=3 we have
s(3)*s(0) = s(3); s(2,1)*s(0) = s(2,1); s(1,1,1)*s(0) = s(1,1,1)
s(2)*s(1) = s(3) + s(2,1) and
s(1,1)*s(1) = s(2,1) + s(1,1,1)
for a total of 3+2+2 = 7 terms.
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MATHEMATICA
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Tr/@ Table[Sum[
Length[LRRule[\[Lambda], \[Mu]]], {\[Lambda],
Partitions[n - i]}, {\[Mu],
If[2 i === n, Join[{\[Lambda]}, lesspartitions[\[Lambda]]],
Partitions[i]]}], {n, 14}, {i, 0, Floor[(n)/2]}]; (* Uses functions defined in the 'Toolbox for symmetric functions', see Links. *)
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CROSSREFS
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Cf. A296625, A296626.
Sequence in context: A147065 A049947 A266822 * A296663 A066345 A026570
Adjacent sequences: A296621 A296622 A296623 * A296625 A296626 A296627
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KEYWORD
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nonn,hard,more
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AUTHOR
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Wouter Meeussen, Dec 17 2017
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STATUS
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approved
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