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A002047 Number of 3 X (2n+1) zero-sum arrays with entries -n,..,0,..,n.
(Formerly M1688 N0666)
8
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 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 so 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

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

Table of n, a(n) for n=0..11.

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.

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]

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

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

Sequence in context: A226773 A119966 A256599 * A126340 A277480 A136639

Adjacent sequences:  A002044 A002045 A002046 * A002048 A002049 A002050

KEYWORD

nonn,nice,more

AUTHOR

N. J. A. Sloane

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

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Last modified August 21 07:08 EDT 2017. Contains 290862 sequences.