A200472 Triangle T(n,k) is the number of ways to assign n people to k unlabeled groups of equal size.

1, 1, 1, 1, 0, 1, 1, 3, 0, 1, 1, 0, 0, 0, 1, 1, 10, 15, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 35, 0, 105, 0, 0, 0, 1, 1, 0, 280, 0, 0, 0, 0, 0, 1, 1, 126, 0, 0, 945, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 462, 5775, 15400, 0, 10395, 0, 0, 0, 0, 0, 1

Offset

1,8

Comments

If k is not a factor of n, T(n,k) = 0. If k is a factor of n, T(n,k) = (n!/k!)/(n/k)!^k. If n is a multiple of k, we may obtain T(n,k) by arranging all n people in an ordered line, which can be done in n! ways. Peel off the first n/k people for "group 1", the next n/k people for "group 2", ..., and the last n/k people for "group k". Since the k groups are actually unlabeled, we must divide n! by k! Also, since the ordering of the n/k people within each of the k groups is not of importance, we must now divide by (n/k)!^k. Therefore, T(n,k) = (n!/k!)/(n/k)!^k.

Also, T(2n,n) provide the sequence consisting of the products of consecutive odd integers.

Links

Table of n, a(n) for n=1..78.

Dennis P. Walsh, Note on assigning n people into k unlabeled groups of equal size

Formula

For k that divide n, T(n,k) = (n!/k!)/((n/k)!)^k; otherwise, T(n,k) = 0.

E.g.f. when k is fixed: (1/k!) sum(j>=1, (x^j/j!)^k ).

E.g.f. for T(n*r,n): exp(x^r/r!).

T(2n,n) = (2n-1)!! = (2n-1)(2n-3)...(3)(1).

Example

Triangle T(n,k) begins
1;
1,   1;
1,   0,    1;
1,   3,    0,     1;
1,   0,    0,     0,   1;
1,  10,   15,     0,   0,     1;
1,   0,    0,     0,   0,     0,  1;
1,  35,    0,   105,   0,     0,  0,  1;
1,   0,  280,     0,   0,     0,  0,  0,  1;
1, 126,    0,     0, 945,     0,  0,  0,  0,  1;
1,   0,    0,     0,   0,     0,  0,  0,  0,  0,  1;
1, 462, 5775, 15400,   0, 10395,  0,  0,  0,  0,  0,  1;
...
T(6,2) = 10 since there are 10 ways to assign 6 people (A,B,C,D,E,F) into 2 groups of size 3. The assignments are {A,B,C}|{D,E,F}, {A,B,D}|{C,E,F}, {A,B,E}|{C,D,F}, {A,B,F}|{C,D,E}, {A,C,D}|{B,E,F}, {A,C,E}|{B,D,F}, {A,C,F}|{B,D,E}, {B,C,D}|{A,E,F}, {B,C,E}|{A,D,F}, and {B,C,F}|{A,D,E}.

Maple

T:= (n, k)-> `if`(modp(n, k)=0, n!/(k!*((n/k)!)^k), 0):
seq(seq(T(n, k), k=1..n), n=1..20);

Mathematica

nn=11; s=Sum[Exp[y x^i/i!]-1, {i, 1, nn}]; Range[0, nn]!CoefficientList[Series[s, {x, 0, nn}], {x, y}]//Grid  (* Geoffrey Critzer, Sep 15 2012 *)

Prog

(PARI)
T(n, k) = if(n%k!=0, 0, (n!/k!)/((n/k)!)^k );
for (n=1, 15, for (k=1, n, print1(T(n, k), ", ")); print());
/* Joerg Arndt, Sep 16 2012 */

Crossrefs

T(2n,n) is A001147(n).

T(3n,n) is A025035(n).

T(4n,n) is A025036(n).

Row sums of T(n,k) provide A038041(n).

A200473 is A200472 with zeros removed.

Sequence in context: A279514 A094675 A307808 * A309887 A317595 A263753

Adjacent sequences: A200469 A200470 A200471 * A200473 A200474 A200475

Keyword

nonn,easy,tabl

Author

Dennis P. Walsh, Nov 18 2011

Status

approved