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A145888
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is the largest entry in the cycle containing 1 (1 <= k <= n).
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1
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1, 1, 1, 2, 1, 3, 6, 2, 4, 12, 24, 6, 10, 20, 60, 120, 24, 36, 60, 120, 360, 720, 120, 168, 252, 420, 840, 2520, 5040, 720, 960, 1344, 2016, 3360, 6720, 20160, 40320, 5040, 6480, 8640, 12096, 18144, 30240, 60480, 181440, 362880, 40320, 50400, 64800
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OFFSET
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1,4
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COMMENTS
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Row sums are the factorials (A000142).
Sum_{k=1..n} k*T(n,k) = A121586(n).
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REFERENCES
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Solution to Problem 1831 by J. W. Grossman. Mathematics Magazine, 83, No. 5, 2010, pp. 392-393.
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LINKS
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FORMULA
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T(n,1)=(n-1)!; T(n,k)=n!/((n-k+1)(n-k+2)) for 2 <= k <= n.
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EXAMPLE
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T(4,3)=4 because we have (132)(4), (13)(24), (123)(4), (13)(2)(4).
Triangle starts:
1;
1, 1;
2, 1, 3;
6, 2, 4, 12;
24, 6, 10, 20, 60;
120, 24, 36, 60, 120, 360;
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MAPLE
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T:=proc(n, k) if k=1 then factorial(n-1) elif k <= n then factorial(n)/((n-k+1)*(n-k+2)) else 0 end if end proc: for n to 10 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form
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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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