A002047 Number of 3 X (2n+1) zero-sum arrays with entries -n,...,0,...,n.

1, 2, 6, 28, 244, 2544, 35600, 659632, 15106128, 425802176, 14409526080, 577386122880

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more

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

Examples

			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.
		

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

• C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
• 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]
• N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136-146.
• Gheorghe Coserea, Solutions for n=4.
• Gheorghe Coserea, Solutions for n=5.
• Gheorghe Coserea, MiniZinc model for generating solutions.
• Diane Donovan, Asha Rao, Elif Üsküplü, and E. Ş. Yazıcı, QC-LDPC Codes from Difference Matrices and Difference Covering Arrays, arXiv:2205.00563 [math.CO], 2022.
• A. Kotzig and P. J. Laufer, When are permutations additive?, Amer. Math. Monthly, 85 (1978), 364-365.
• 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]
• Wikipedia, Hexagonal chess - Gliński's hexagonal chess

Crossrefs

Cf. A014552. A diagonal of the triangle in A260333.

Cf. A309260, A309746.

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