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%I M1688 N0666 #93 Mar 23 2024 23:30:01
%S 1,2,6,28,244,2544,35600,659632,15106128,425802176,14409526080,
%T 577386122880
%N Number of 3 X (2n+1) zero-sum arrays with entries -n,...,0,...,n.
%C 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
%C 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]
%C 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
%C Also the number of sigma-permutations of length 2n+1 [Kotzig-Laufer]. - _N. J. A. Sloane_, Jul 27 2015
%C 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
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="https://arxiv.org/abs/math/0506334">On the X-rays of permutations</a>, arXiv:math/0506334 [math.CO], 2005.
%H B. T. Bennett and R. B. Potts, <a href="https://doi.org/10.1017/S144678870000505X">Arrays and brooks</a>, J. Austral. Math. Soc., 7 (1967), 23-31.
%H B. T. Bennett and R. B. Potts, <a href="/A002047/a002047_1.pdf">Arrays and brooks</a>, J. Austral. Math. Soc., 7 (1967), 23-31. [Annotated scanned copy]
%H N. J. Cavenagh and I. M. Wanless, <a href="http://dx.doi.org/10.1016/j.dam.2009.09.006">On the number of transversals in Cayley tables of cyclic groups</a>, Disc. Appl. Math. 158 (2010), 136-146.
%H Gheorghe Coserea, <a href="/A002047/a002047.txt">Solutions for n=4</a>.
%H Gheorghe Coserea, <a href="/A002047/a002047_1.txt">Solutions for n=5</a>.
%H Gheorghe Coserea, <a href="/A002047/a002047.mzn.txt">MiniZinc model for generating solutions</a>.
%H Diane Donovan, Asha Rao, Elif Üsküplü, and E. Ş. Yazıcı, <a href="https://arxiv.org/abs/2205.00563">QC-LDPC Codes from Difference Matrices and Difference Covering Arrays</a>, arXiv:2205.00563 [math.CO], 2022.
%H A. Kotzig and P. J. Laufer, <a href="http://www.jstor.org/stable/2321345">When are permutations additive?</a>, Amer. Math. Monthly, 85 (1978), 364-365.
%H A. Kotzig and P. J. Laufer, <a href="/A002047/a002047.pdf">When are permutations additive?</a>, 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]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hexagonal_chess#Gli%C5%84ski's_hexagonal_chess">Hexagonal chess - Gliński's hexagonal chess</a>
%e a(2) = 6 corresponds to
%e ..O.X.X.......X.X.O.......O.X.X.......X.O.X.......X.O.X.......X.X.O
%e .X.X.O.X.....X.O.X.X.....X.X.X.O.....X.X.X.O.....O.X.X.X.....O.X.X.X
%e 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
%e .O.X.X.X.....X.X.X.O.....X.X.X.O.....X.O.X.X.....X.X.O.X.....O.X.X.X
%e ..X.O.X.......X.O.X.......O.X.X.......X.X.O.......O.X.X.......X.X.O
%e The bijection with a pair of the 3 X (2n+1) zero-sum arrays:
%e n=1, a(1)=2 corresponds to
%e 3 4 2 3 2 4
%e and mirror image 4 2 3 2 4 3
%e element 2 3 4 -(2n+1) --> -1 0 1
%e position, left element 3 1 2 -( n+1) --> 1 -1 0
%e position in mirror 2 3 1 -( n+1) --> 0 1 -1
%e ------- -------
%e sum of column 7 7 7 -(4n+3) 0 0 0
%e Swapping rows 2,3 yields the other 3 X 3 zero sum array.
%e n=2, a(2)=6 an example and its mirror, so 2 of the 6 solutions:
%e 5 6 7 3 4 5 3 6 4 7
%e mirror image 7 4 6 3 5 4 3 7 6 5
%e 3 4 5 6 7 -(2n+1) --> -2 -1 0 1 2
%e 4 5 1 2 3 -( n+1) --> 1 2 -2 -1 0
%e 4 2 5 3 1 -( n+1) --> 1 -1 2 0 -2
%e -------------- --------------
%e 11 11 11 11 11 -(4n+3) --> 0 0 0 0 0
%e Swapping rows 2,3 yields the other 3 X 5 zero sum array.
%Y Cf. A014552. A diagonal of the triangle in A260333.
%Y Cf. A309260, A309746.
%K nonn,nice,more
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Alex Fink (a00(AT)shaw.ca), Mar 16 2005
%E a(10) and a(11) from _Ian Wanless_, Jul 30 2010, from the Cavenagh-Wanless paper