A137268 -id:A137268 - cp's OEIS frontend

A152684 a(n) is the number of top-down sequences (F_1, F_2, ..., F_n) whereas each F_i is a labeled forest on n nodes, containing i directed rooted trees. F_(i+1) is proper subset of F_i.

1, 2, 18, 384, 15000, 933120, 84707280, 10569646080, 1735643790720, 362880000000000, 94121726392108800, 29658516531078758400, 11159820050604594969600, 4942478402320838374195200, 2544989406021562500000000000, 1507645899890367707813511168000

Offset: 1

Fabian Nedic, Dec 10 2008

nonn

			a(1) = 1^(1-2)*(1!) = 1.
a(2) = 2^(2-2)*(2!) = 2.
a(3) = 3^(3-2)*(3!) = 18.

Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill 2007, Page 276.

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A061711, A066319, A137268.

[Factorial(n-1)*n^(n-1): n in [1..20]]; // G. C. Greubel, Nov 28 2022

a:= proc(n) option remember; `if`(n=1, 1,
      a(n-1)*(n/(n-1))^(n-3)*n^2)
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, May 16 2013

Table[n^(n - 1) (n - 1)!, {n, 1, 16}]  (* Geoffrey Critzer, May 10 2013 *)

[factorial(n-1)*n^(n-1) for n in range(1,21)] # G. C. Greubel, Nov 28 2022

a(n) = n^(n-2)*(n!).

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

Original entry on oeis.org

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

Ralf Stephan, Feb 26 2005

			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;

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

Cf. A000079, A000244, A000302, A137268,

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

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

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

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)

cp's OEIS Frontend

A152684 a(n) is the number of top-down sequences (F_1, F_2, ..., F_n) whereas each F_i is a labeled forest on n nodes, containing i directed rooted trees. F_(i+1) is proper subset of F_i.

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