A152883 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is an excedance (n >= 2, 1 <= k <= n-1). An excedance of a permutation p is a value j such that p(j) > j.

1, 4, 2, 18, 12, 6, 96, 72, 48, 24, 600, 480, 360, 240, 120, 4320, 3600, 2880, 2160, 1440, 720, 35280, 30240, 25200, 20160, 15120, 10080, 5040, 322560, 282240, 241920, 201600, 161280, 120960, 80640, 40320, 3265920, 2903040, 2540160, 2177280, 1814400, 1451520, 1088640, 725760, 362880

Offset: 2

Author

Emeric Deutsch, Jan 13 2009

Keywords

• nonn
• tabl

Comments

Sum of entries in row n = n!*(n-1)/2 = A001286(n) (the Lah numbers).

T(n,n-1) = (n-1)! (A000142).

Examples

			T(4,3) = 6 because the permutations of {1,2,3,4} in which 3 is an excedance are 1243, 1342, 3142, 2143, 2341 and 3241.
Triangle starts:
    1;
    4,   2;
   18,  12,   6;
   96,  72,  48,  24;
  600, 480, 360, 240, 120;
		

Crossrefs

Cf. A000142, A001286.

Programs

Maple

T := proc (n, k) options operator, arrow: factorial(n-1)*(n-k) end proc: for n from 2 to 10 do seq(T(n, k), k = 1 .. n-1) end do;

Formula

T(n,k) = (n-1)!*(n-k) (n >= 2, 1 <= k <= n-1). [Proof: n-k choices for p(k) and (n-1)! choices for the remaining entries of p.]