A104001 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns starting with fixed m, 2

2, 4, 6, 8, 18, 24, 16, 54, 96, 120, 32, 162, 384, 600, 720, 64, 486, 1536, 3000, 4320, 5040, 128, 1458, 6144, 15000, 25920, 35280, 40320, 256, 4374, 24576, 75000, 155520, 246960, 322560, 362880, 512, 13122, 98304, 375000, 933120, 1728720, 2580480, 3265920, 3628800

Offset: 3

Author

Ralf Stephan, Feb 26 2005

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
    2;
    4,    6;
    8,   18,   24;
   16,   54,   96,   120;
   32,  162,  384,   600,   720;
   64,  486, 1536,  3000,  4320,  5040;
  128, 1458, 6144, 15000, 25920, 35280, 40320;
		

Links

• G. C. Greubel, Rows n = 3..50 of the triangle, flattened
• T. Mansour, Permutations containing and avoiding certain patterns

Crossrefs

Cf. A000079, A000244, A000302, A137268,

Programs

Magma

[Factorial(k-1)*(k-1)^(n-k): k in [3..n], n in [3..15]]; // G. C. Greubel, Nov 29 2022

Mathematica

Table[(k-1)!*(k-1)^(n-k), {n,3,15}, {k,3,n}]//Flatten (* G. C. Greubel, Nov 29 2022 *)

SageMath

def A104001(n,k): return factorial(k-1)*(k-1)^(n-k)
flatten([[A104001(n,k) for k in range(3,n+1)] for n in range(3,16)]) # G. C. Greubel, Nov 29 2022

Formula

T(n, k) = (k-2)! * (k-1)^(n+1-k).

From G. C. Greubel, Nov 29 2022: (Start)

T(n, 3) = A000079(n-2).

T(n, 4) = 6*A000244(n-4).

T(n, 5) = 4!*A000302(n-5).

T(2*n-3, n) = A152684(n-1). (End)