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A248090
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Square array read by antidiagonals: T(n,k) is the number of k-edge colored trees on vertex set [n] (n>=2, k>=2).
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0
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2, 3, 6, 4, 18, 24, 5, 36, 168, 120, 6, 60, 528, 2160, 720, 7, 90, 1200, 10920, 35640, 5040, 8, 126, 2280, 34200, 293760, 720720, 40320, 9, 168, 3864, 82800, 1275120, 9767520, 17297280, 362880, 10, 216, 6048, 170520, 3946320, 58968000, 387636480, 481178880, 3628800
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OFFSET
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2,1
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COMMENTS
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999, Exercise 5.28, pp. 81, 124.
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LINKS
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FORMULA
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T(n,k) = k(n-2)!*binomial((k-1)n, n-2).
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EXAMPLE
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T(3,3) = 18; indeed, a 3-vertex tree ABC can be labeled in 6 ways and for each labeled tree the 2 edges can be colored in 3 ways (a and b, a and c, b and c).
The antidiagonals start:
2;
3, 6;
4, 18, 24;
5, 36, 168, 120;
6, 60, 528, 2160, 720;
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MAPLE
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T := proc(n, k) options operator, arrow: k*factorial(n-2)*binomial((k-1)*n, n-2) end proc: seq(seq(T(i, j-i), i = 2 .. j-2), j = 4 .. 12); # the command T(n, k) yields T(n, k).
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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