A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

1, 2, 1, 6, 3, 2, 24, 12, 8, 6, 120, 60, 40, 30, 24, 720, 360, 240, 180, 144, 120, 5040, 2520, 1680, 1260, 1008, 840, 720, 40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040, 362880, 181440, 120960, 90720, 72576, 60480, 51840, 45360, 40320, 3628800, 1814400, 1209600, 907200, 725760, 604800, 518400, 453600, 403200, 362880

Offset: 1

Author

Amarnath Murthy, Apr 29 2004

Keywords

• nonn
• tabl

Comments

The sum of the rows gives A000254 (Stirling numbers of first kind). The first column and the leading diagonal are factorials given by A000142 with offsets of 0 and 1.

T(n,k) is the number of length k cycles in all permutations of {1..n}.

Second diagonal gives A001048(n). - Anton Zakharov, Oct 24 2016

T(n,k) is the number of permutations of [n] with all elements of [k] in a single cycle. To prove this result, let m denote the length of the cycle containing {1,..,k}. Letting m run from k to n, we obtain T(n,k) = Sum_{m=k..n} (C(n-k,m-k)*(m-1)!*(n-m)!) = n!/k. See an example below. - Dennis P. Walsh, May 24 2020

Examples

			Triangle begins as:
      1;
      2,     1;
      6,     3,     2;
     24,    12,     8,     6;
    120,    60,    40,    30,   24;
    720,   360,   240,   180,  144,  120;
   5040,  2520,  1680,  1260, 1008,  840,  720;
  40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040;
  ...
T(4,2) counts the 12 permutations of [4] with elements 1 and 2 in the same cycle, namely, (1 2)(3 4), (1 2)(3)(4), (1 2 3)(4), (1 3 2)(4), (1 2 4)(3), (1 4 2)(3), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A000142, A000254, A001563, A001710, A002301.

Cf. A061579, A094307, A110468, A133799.

Cf. A129825. - Johannes W. Meijer, Jun 18 2009

Programs

Maple

seq(seq(n!/k, k=1..n), n=1..10);

Mathematica

Table[n!/k, {n,10}, {k,n}]//Flatten
Table[n!/Range[n], {n,10}]//Flatten (* Harvey P. Dale, Mar 12 2016 *)

Formula

E.g.f. for column k: x^k/(k*(1-x)).

T(n,k)*k = n*n! = A001563(n).

Extensions

More terms from Philippe Deléham, Jun 11 2005