|
| |
|
|
A002047
|
|
Number of 3 X (2n+1) zero-sum arrays with entries -n,...,0,...,n.
(Formerly M1688 N0666)
|
|
11
|
|
|
|
1, 2, 6, 28, 244, 2544, 35600, 659632, 15106128, 425802176, 14409526080, 577386122880
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
This can be interpreted as the number of ways to choose 2n+1 cells in a hexagonal grid of side n+1 such that no two are in the same row or left diagonal or right diagonal. - Alex Fink (a00(AT)shaw.ca), Mar 16 2005
Also the number of transversals of a partial Latin square L of order 2n+1 in which L_{ij} = i+j if n+1 < i+j < 3n+3 and L_{ij} is empty otherwise. [Cavenagh-Wanless]
Also the number of arrangements of the numbers n+1, n+1, ..., 3n+1, 3n+1 such that there are n numbers between the pair of n+1's, ..., 3n numbers between the pair of 3n+1's. For each of these arrangements and its mirror image, there is a bijection with a pair of the 3 X (2n+1) zero-sum arrays. - Stephen J Scattergood, Jul 19 2013
Also the number of sigma-permutations of length 2n+1 [Kotzig-Laufer]. - N. J. A. Sloane, Jul 27 2015
An (m,2n+1)-zero-sum array is an m X (2n+1) matrix whose m rows are permutations of the 2n+1 integers -n..n, the sum of each column is zero and the first row of the matrix is -n,-n+1,...,0,...,n-1,n. - Gheorghe Coserea, Dec 29 2016
a(n-1) is also number of ways of placing 2*n-1 nonattacking rooks on a hexagonal board with edge-length n in Glinski's hexagonal chess. - Vaclav Kotesovec, Aug 15 2019
|
|
|
REFERENCES
|
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31.
B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31. [Annotated scanned copy]
A. Kotzig and P. J. Laufer, When are permutations additive?, Amer. Math. Monthly, 85 (1978), 364-365. [Annotated by C. L. Mallows, scanned copy, together with letter from C. L. Mallows and N. J. A. Sloane to A. Kotzig, Jul 25 1978]
|
|
|
EXAMPLE
|
a(2) = 6 corresponds to
..O.X.X.......X.X.O.......O.X.X.......X.O.X.......X.O.X.......X.X.O
.X.X.O.X.....X.O.X.X.....X.X.X.O.....X.X.X.O.....O.X.X.X.....O.X.X.X
X.X.X.X.O...O.X.X.X.X...X.O.X.X.X...O.X.X.X.X...X.X.X.X.O...X.X.X.O.X
.O.X.X.X.....X.X.X.O.....X.X.X.O.....X.O.X.X.....X.X.O.X.....O.X.X.X
..X.O.X.......X.O.X.......O.X.X.......X.X.O.......O.X.X.......X.X.O
The bijection with a pair of the 3 X (2n+1) zero-sum arrays:
n=1, a(1)=2 corresponds to
3 4 2 3 2 4
and mirror image 4 2 3 2 4 3
element 2 3 4 -(2n+1) --> -1 0 1
position, left element 3 1 2 -( n+1) --> 1 -1 0
position in mirror 2 3 1 -( n+1) --> 0 1 -1
------- -------
sum of column 7 7 7 -(4n+3) 0 0 0
Swapping rows 2,3 yields the other 3 X 3 zero sum array.
n=2, a(2)=6 an example and its mirror, so 2 of the 6 solutions:
5 6 7 3 4 5 3 6 4 7
mirror image 7 4 6 3 5 4 3 7 6 5
3 4 5 6 7 -(2n+1) --> -2 -1 0 1 2
4 5 1 2 3 -( n+1) --> 1 2 -2 -1 0
4 2 5 3 1 -( n+1) --> 1 -1 2 0 -2
-------------- --------------
11 11 11 11 11 -(4n+3) --> 0 0 0 0 0
Swapping rows 2,3 yields the other 3 X 5 zero sum array.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
More terms from Alex Fink (a00(AT)shaw.ca), Mar 16 2005
a(10) and a(11) from Ian Wanless, Jul 30 2010, from the Cavenagh-Wanless paper
|
|
|
STATUS
|
approved
|
| |
|
|
