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THERE ARE ASYMPTOTICALLY THE SAME NUMBER OF LATIN SQUARES OF EACH PARITY

Part of: Designs and configurations

Published online by Cambridge University Press:  21 July 2016

NICHOLAS J. CAVENAGH  and
IAN M. WANLESS
NICHOLAS J. CAVENAGH
Affiliation:
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand email [email protected]
IAN M. WANLESS*
Affiliation:
School of Mathematical Sciences, Monash University, Victoria 3800, Australia email [email protected]
*
✉[email protected]
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Abstract

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A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order $n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow \infty$.

Keywords

Latin squareparityAlon–Tarsi conjecturerow cycle

MSC classification

Primary: 05B15: Orthogonal arrays, Latin squares, Room squares
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 187 - 194
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Alon, N. and Tarsi, M., ‘Colorings and orientations of graphs’, Combinatorica 12 (1992), 125–134.CrossRefGoogle Scholar
Alpoge, L., ‘Square-root cancellation for the signs of Latin squares’, Combinatorica, to appear, doi:10.1007/s00493-015-3373-7.Google Scholar
Cavenagh, N. J., Greenhill, C. and Wanless, I. M., ‘The cycle structure of two rows in a random Latin square’, Rand. Struct. Algorithms 33 (2008), 286–309.Google Scholar
Huang, R. and Rota, G.-C., ‘On the relations of various conjectures on Latin squares and straightening coefficients’, Discrete Math. 128 (1994), 225–236.Google Scholar
Kaski, P., Medeiros, A. D. S., Östergård, P. R. J. and Wanless, I. M., ‘Switching in one-factorisations of complete graphs’, Electron. J. Combin. 21(2) (2014), #P2.49, 24 pages.CrossRefGoogle Scholar
Kotlar, D., ‘Parity types, cycle structures and autotopisms of Latin squares’, Electron. J. Combin. 19(3) (2012), #P10, 17 pages.CrossRefGoogle Scholar
Kotlar, D., ‘On extensions of the Alon–Tarsi Latin square conjecture’, Electron. J. Combin. 19(4) (2012), #P7, 10 pages.CrossRefGoogle Scholar
Kumar, S. and Landsberg, J. M., ‘Connections between conjectures of Alon–Tarsi, Hadamard–Howe, and integrals over the special unitary group’, Discrete Math. 338 (2015), 1232–1238.Google Scholar
Lefevre, J. G., Donovan, D. M., Grannell, M. J. and Griggs, T. S., ‘A constraint on the biembedding of Latin squares’, European J. Combin. 30 (2009), 380–386.CrossRefGoogle Scholar
McKay, B. D. and Wanless, I. M., ‘On the number of Latin squares’, Ann. Comb. 9 (2005), 335–344.Google Scholar
Stones, D. S., ‘Formulae for the Alon–Tarsi conjecture’, SIAM J. Discrete Math. 26 (2012), 65–70.CrossRefGoogle Scholar
Stones, D. S. and Wanless, I. M., ‘How not to prove the Alon–Tarsi conjecture’, Nagoya Math. J. 205 (2012), 1–24.CrossRefGoogle Scholar
Wanless, I. M., ‘Cycle switches in Latin squares’, Graphs Combin. 20 (2004), 545–570.CrossRefGoogle Scholar
Wilf, H. S., Generating Functionology, 2nd edn (Academic Press, San Diego, CA, 1994).Google Scholar
Zappa, P., ‘The Cayley determinant of the determinant tensor and the Alon–Tarsi conjecture’, Adv. Appl. Math. 13 (1997), 31–44.CrossRefGoogle Scholar