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Search: id:A138785
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| A138785 |
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Triangle read by rows: T(n,k) is the number of hook lengths equal to k among all hook lengths of all partitions of n (1<=k<=n). |
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1
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| 1, 2, 2, 4, 2, 3, 7, 6, 3, 4, 12, 8, 6, 4, 5, 19, 16, 12, 8, 5, 6, 30, 22, 18, 12, 10, 6, 7, 45, 38, 27, 24, 15, 12, 7, 8, 67, 52, 45, 32, 25, 18, 14, 8, 9, 97, 82, 63, 52, 40, 30, 21, 16, 9, 10, 139, 112, 93, 72, 60, 42, 35, 24, 18, 10, 11, 195, 166, 135, 112, 85, 72, 49, 40, 27, 20
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums yield A066186(n)=n*A000041(n).
T(n,1)=A000070.
Sum(k*T(n,k),k=1..n)=A066183(n).
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REFERENCES
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R. Bacher and L. Manivel, Hooks and powers of parts in partitions, Sem. Lotharingien de Combinatoire, 47, 2002, B47d.
Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 24).
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FORMULA
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G.f.=Sum(k*t^k*x^k/[(1-x^k)*Product(1-x^m,m=1..infinity)],k=1..infinity). T(n,k)=k*A066633(n,k).
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EXAMPLE
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T(4,2)=6 because for the partitions (4), (3,1), (2,2), (2,1,1), (1,1,1,1) of n=4 the hook length multi-sets are {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1}, respectively, containing altogether six 2's.
Triangle starts:
1;
2,2;
4,2,3;
7,6,3,4;
12,8,6,4,5;
19,16,12,8,5,6;
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MAPLE
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g:=sum(k*x^k*t^k/((1-x^k)*(product(1-x^m, m=1..50))), k=1..50): gser:= simplify(series(g, x=0, 15)): for n to 12 do P[n]:= sort(coeff(gser, x, n)) end do: for n to 12 do seq(coeff(P[n], t, j), j=1..n) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A000041, A066186, A000070, A066183, A066633.
Sequence in context: A043262 A130860 A161535 this_sequence A131817 A138232 A102641
Adjacent sequences: A138782 A138783 A138784 this_sequence A138786 A138787 A138788
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 16 2008
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