A199673 Number of ways to form k labeled groups, each with a distinct leader, using n people. Triangle T(n,k) = n!*k^(n-k)/(n-k)! for 1 <= k <= n.

1, 2, 2, 3, 12, 6, 4, 48, 72, 24, 5, 160, 540, 480, 120, 6, 480, 3240, 5760, 3600, 720, 7, 1344, 17010, 53760, 63000, 30240, 5040, 8, 3584, 81648, 430080, 840000, 725760, 282240, 40320, 9, 9216, 367416, 3096576, 9450000, 13063680, 8890560, 2903040, 362880

Offset

1,2

Comments

T(n,1)=n since there are n choices for the leader of the single group. Also, T(n,n)=n! since each of the n groups consist solely of a leader and there are n! ways to assign the n people to the n labeled groups.

In general, T(n,k) = n!*k^(n-k)/(n-k)! since there are n!/(n-k)! ways to assign leaders to the k labeled groups and there are k^(n-k) ways to map the remaining (n-k) people to the k groups.

T(n,k) is the number of functions of [n] to an arbitrary k-subset of [n], where each of the k target values is used at least once.

The number of ways to distribute n different toys among k girls and k boys to that each girl gets exactly one toy. - Dennis P. Walsh, Sep 10 2012

Links

Joerg Arndt, Table of n, a(n) for n = 1..561

George Kesidis, Takis Konstantopoulos, and Michael A. Zazanis, Age of information without service preemption, arXiv:2104.08050 [cs.PF], 2021.

Dennis P. Walsh, Assigning people into labeled groups with leaders

Dennis P. Walsh, Toy Story 2

Formula

T(n,k) = n!*k^(n-k)/(n-k)! = k!*k^(n-k)*binomial(n,k) for 1 <= k <= n.

E.g.f.: (x*e^x)^k,for fixed k.

T(n,k1+k2) = Sum_{j=0..n} binomial(n,j)*T(j,k1)*T(n-j,k2).

T(n,1) = A000027(n);

T(n,2) = A001815(n);

T(n,3) = A052791(n);

Sum_{k=1..n} T(n,k) = A006153(n).

T(n,n) = A000142(n) = n!. - Dennis P. Walsh, Sep 10 2012

Example

T(3,2)=12 since there are 12 ways to form group 1 and group 2, both with leaders, using people p1, p2, and p3, as illustrated below. The leader will be denoted Lj if person pj is designated the leader of the group.
Group 1   Group 2
{L1,p2}   {L3}
{L1,p3}   {L2}
{L1}      {L2,p3}
{L1}      {p2,L3}
{L2,p1}   {L3}
{L2,p3}   {L1}
{L2}      {L1,p3}
{L2}      {p1,L3}
{L3,p2}   {L1}
{L3,p1}   {L2}
{L3}      {L1,p2}
{L3}      {p1,L2}
First rows of triangle T(n,k):
  1;
  2,    2;
  3,   12,      6;
  4,   48,     72,      24;
  5,  160,    540,     480,     120;
  6,  480,   3240,    5760,    3600,      720;
  7, 1344,  17010,   53760,   63000,    30240,    5040;
  8, 3584,  81648,  430080,  840000,   725760,  282240,   40320;
  9, 9216, 367416, 3096576, 9450000, 13063680, 8890560, 2903040, 362880;

Maple

seq(seq(n!*k^(n-k)/(n-k)!, k=1..n), n=1..9);

Mathematica

nn = 10; a = y x Exp[x]; f[list_] := Select[list, # > 0 &]; Drop[Map[f, Range[0, nn]! CoefficientList[Series[1/(1 - a) , {x, 0, nn}], {x, y}]], 1] // Flatten  (* Geoffrey Critzer, Jan 21 2012 *)

Prog

(Magma) [Factorial(n)*k^(n-k)/Factorial(n-k): k in [1..n], n in [1..9]];  // Bruno Berselli, Nov 09 2011
(PARI)
T(n, k)=n!*k^(n-k)/(n-k)!;
/* print triangle: */
for (n=1, 15, for (k=1, n, print1(T(n, k), ", ")); print() );
/* Joerg Arndt, Sep 21 2012 */

Crossrefs

Sequence in context: A156136 A134243 A182779 * A335942 A240133 A293445

Adjacent sequences: A199670 A199671 A199672 * A199674 A199675 A199676

Keyword

nonn,easy,tabl

Author

Dennis P. Walsh, Nov 08 2011

Status

approved