Christian G.Bower bowerc at usa.net
Thu Feb 14 17:59:40 CET 2002

```ID Number: A047996
Sequence:  1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,3,4,3,1,1,1,1,
3,5,5,3,1,1,1,1,4,7,10,7,4,1,1,1,1,4,10,14,14,10,4,1,1,1,1,
5,12,22,26,22,12,5,1,1,1,1,5,15,30,42,42,30,15,5,1,1,1,1,6,
19,43,66,80,66,43,19,6,1,1,1,1,6,22
Name:      Triangle of circular binomial coefficients T(n,k), 0<=k<=n.
Comments:  T(n,k)=number of necklaces with k black beads, n-k white beads.
References D. E. Knuth, Computer science and its relation to mathematics,
Amer.
Math. Monthly, 81 (1974), 323-343.
H. S. Wilf, personal communication, Nov., 1990.
Links:     Index entries for sequences related to necklaces
Formula:   T(n,k)=(1/n) * Sum_{d | (n,k)} phi(d)*binomial(n/d,k/d).
Example:   1; 1,1; 1,1,1; 1,1,1,1; 1,1,2,1,1; 1,1,2,2,1,1; 1,1,3,4,3,1,1; ...
See also:  Row sums: A000031. Columns 0-12: A000012, A000012, A004526,
A007997(n-3), A008610, A008646, A032191-A032197.
Cf. A051168, A052307, A052311-A052313.
Keywords:  nonn,tabl,easy,nice
Offset:    0
Author(s): njas

T(n,k) corresponds to n=n, p=k, except that in the partition case,
T(n,0)=0 while in the necklace case T(n,0)=1.

This table includes odd values of n and k which the correspondent
excluded.

Christian

"N. J. A. Sloane" <njas at research.att.com> wrote:
> Anyone want to help this correspondent?  -  njas
>
>
>
> >>From mahmoud.bekheit at eee.strath.ac.uk  Thu Feb 14 10:12:33 2002
> >>Delivered-To: njas at research.att.com
> >>From: "Mahmoud Bekheit" <mahmoud.bekheit at eee.strath.ac.uk>
> >>To: <njas at research.att.com>
> >>Subject: Help: Partitions of a number into summand parts & Non-equivalent
under rotation
...
> >>Dear Prof. Sloane
> >>
> >>I'm interested in generating all partitions of an integer number n into
> >>parts p, each partition should be permutated to five all possible prtts
> >>permutations. This permutation should satisfy the "non-equivalent under
> >>rotation", like Necklaces.
...
> >>sample results are:
> >>
> >>p          2        4        6        8        10
> >>===================================
> >>n
> >>2          1
> >>4          2        1
> >>6          3        3        1
> >>8          4        10      4        1
> >>10        5        22      22      5        1
> >>12        6        43      80      43      6      1
> >>14        7        73      217    217    73    7     1
> >>16        8        116    504    810    504  116        1
> >>
> >>My question to you
> >>Is there any definition of this sequence(non equivalent under rotation
for
> >>set of p parts)
> >>
> >>Thanks
> >>Mahmoud Bekheit
> >>University of Strathclyde
> >>Department of Electronic and Electrical Engineering
> >>Royal College Building
> >>204 George Street
> >>Glasgow G1 1XW
> >>Scotland, U.K
> >>
> >>Tel: 0141 548 2544
> >>Int.: +44 141 548 2544
> >>Fax: 0141 552 2487
> >>E-mail: mahmoud.bekheit at eee.strath.ac.uk
> >>

```