A145888 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).

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

Offset: 1

Author

Emeric Deutsch, Nov 10 2008

Keywords

• nonn
• tabl

Comments

Row sums are the factorials (A000142).

T(n,n) = n!/2 = A001710(n) (n>=2).

Sum_{k=1..n} k*T(n,k) = A121586(n).

Examples

			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;
		

References

• Solution to Problem 1831 by J. W. Grossman. Mathematics Magazine, 83, No. 5, 2010, pp. 392-393.

Links

• J. W. Grossman, Solution to Problem 1831 proposed by Emeric Deutsch, Mathematics Magazine, 83, No. 5, 2010, pp. 392-393.

Crossrefs

Cf. A000142, A001710, A121586.

Programs

Maple

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

Formula

T(n,1)=(n-1)!; T(n,k)=n!/((n-k+1)(n-k+2)) for 2 <= k <= n.