A104001 - OEIS

%I #7 Nov 29 2022 04:04:28

%S 2,4,6,8,18,24,16,54,96,120,32,162,384,600,720,64,486,1536,3000,4320,

%T 5040,128,1458,6144,15000,25920,35280,40320,256,4374,24576,75000,

%U 155520,246960,322560,362880,512,13122,98304,375000,933120,1728720,2580480,3265920,3628800

%N Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns starting with fixed m, 2<k<=n, 1<=m<=k.

%H G. C. Greubel, <a href="/A104001/b104001.txt">Rows n = 3..50 of the triangle, flattened</a>

%H T. Mansour, <a href="http://arXiv.org/abs/math.CO/9911243">Permutations containing and avoiding certain patterns</a>

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

%F From _G. C. Greubel_, Nov 29 2022: (Start)

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

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

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

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

%e Triangle begins as:

%e     2;

%e     4,    6;

%e     8,   18,   24;

%e    16,   54,   96,   120;

%e    32,  162,  384,   600,   720;

%e    64,  486, 1536,  3000,  4320,  5040;

%e   128, 1458, 6144, 15000, 25920, 35280, 40320;

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

%o (Magma) [Factorial(k-1)*(k-1)^(n-k): k in [3..n], n in [3..15]]; // _G. C. Greubel_, Nov 29 2022

%o (SageMath)

%o def A104001(n,k): return factorial(k-1)*(k-1)^(n-k)

%o flatten([[A104001(n,k) for k in range(3,n+1)] for n in range(3,16)]) # _G. C. Greubel_, Nov 29 2022

%Y Cf. A000079, A000244, A000302, A137268,

%K nonn,tabl

%O 3,1

%A _Ralf Stephan_, Feb 26 2005