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A200472 Triangle T(n,k) is the number of ways to assign n people to k unlabeled groups of equal size. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified March 28 19:51 EDT 2020. Contains 333103 sequences. (Running on oeis4.)