{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,1]],"date-time":"2025-06-01T04:10:44Z","timestamp":1748751044520,"version":"3.41.0"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2015,11,14]],"date-time":"2015-11-14T00:00:00Z","timestamp":1447459200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (CN)","doi-asserted-by":"publisher","award":["61170301"],"award-info":[{"award-number":["61170301"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (CN)","doi-asserted-by":"publisher","award":["61170301"],"award-info":[{"award-number":["61170301"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China (CN)","doi-asserted-by":"publisher","award":["11450110409"],"award-info":[{"award-number":["11450110409"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2016,5]]},"DOI":"10.1007\/s00373-015-1643-1","type":"journal-article","created":{"date-parts":[[2015,11,14]],"date-time":"2015-11-14T06:38:30Z","timestamp":1447483110000},"page":"1187-1202","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Computing the Number of Latin Rectangles"],"prefix":"10.1007","volume":"32","author":[{"given":"Rebecca J.","family":"Stones","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sheng","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoguang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,11,14]]},"reference":[{"key":"1643_CR1","doi-asserted-by":"crossref","first-page":"632","DOI":"10.2307\/2319697","volume":"82","author":"R Alter","year":"1975","unstructured":"Alter, R.: How many Latin squares are there? Amer. Math. Monthly 82, 632\u2013634 (1975)","journal-title":"Amer. Math. Monthly"},{"key":"1643_CR2","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/S0195-6698(80)80015-1","volume":"1","author":"KB Athreya","year":"1980","unstructured":"Athreya, K.B., Pranesachar, C.R., Singhi, N.M.: On the number of Latin rectangles and chromatic polynomial of $${L}({K}_{r, s})$$ L ( K r , s ) . European J. Combin. 1, 9\u201317 (1980)","journal-title":"European J. Combin."},{"key":"1643_CR3","first-page":"463","volume":"54","author":"KP Bogart","year":"1976","unstructured":"Bogart, K.P., Longyear, J.Q.: Counting $$3$$ 3 by $$n$$ n Latin rectangles. Proc. Amer. Math. Soc. 54, 463\u2013467 (1976)","journal-title":"Proc. Amer. Math. Soc."},{"issue":"1","key":"1643_CR4","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s00493-013-2809-1","volume":"13","author":"J Browning","year":"2013","unstructured":"Browning, J., Stones, D.S., Wanless, I.M.: Bounds on the number of autotopisms and subsquares of a Latin square. Combinatorica 13(1), 11\u201322 (2013)","journal-title":"Combinatorica"},{"issue":"1","key":"1643_CR5","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1090\/S0002-9939-1953-0053137-4","volume":"4","author":"L Carlitz","year":"1953","unstructured":"Carlitz, L.: Congruences connected with three-line Latin rectangles. Proc. Amer. Math. Soc. 4(1), 9\u201311 (1953)","journal-title":"Proc. Amer. Math. Soc."},{"key":"1643_CR6","unstructured":"Doyle P.G.: The number of Latin rectangles. arXiv:math\/0703896v1 [math.CO], 15 pp"},{"issue":"1","key":"1643_CR7","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1006\/aima.1997.1623","volume":"128","author":"AA Drisko","year":"1997","unstructured":"Drisko, A.A.: On the number of even and odd Latin squares of order $$p+1$$ p + 1 . Adv. Math. 128(1), 20\u201335 (1997)","journal-title":"Adv. Math."},{"key":"1643_CR8","doi-asserted-by":"crossref","unstructured":"Drisko A.A.: Proof of the Alon-Tarsi Conjecture for $$n=2^r p$$ n = 2 r p , Electron. J. Combin. 5\u00a0(1), R28","DOI":"10.37236\/1366"},{"issue":"8","key":"1643_CR9","doi-asserted-by":"crossref","first-page":"458","DOI":"10.2307\/2305657","volume":"52","author":"AL Dulmage","year":"1945","unstructured":"Dulmage, A.L.: E650. Amer. Math. Monthly 52(8), 458 (1945)","journal-title":"Amer. Math. Monthly"},{"key":"1643_CR10","first-page":"279","volume":"14","author":"AL Dulmage","year":"1975","unstructured":"Dulmage, A.L., McMaster, G.E.: A formula for counting three-line Latin rectangles. Congr. Numer. 14, 279\u2013289 (1975)","journal-title":"Congr. Numer."},{"key":"1643_CR11","doi-asserted-by":"crossref","first-page":"230","DOI":"10.2307\/2371834","volume":"68","author":"P Erd\u0151s","year":"1946","unstructured":"Erd\u0151s, P., Kaplansky, I.: The asymptotic number of Latin rectangles. Amer. J. Math. 68, 230\u2013236 (1946)","journal-title":"Amer. J. Math."},{"key":"1643_CR12","first-page":"239","volume":"103","author":"RM Falc\u00f3n","year":"2012","unstructured":"Falc\u00f3n, R.M.: Cycle structures of autotopisms of the Latin squares of order up to $$11$$ 11 . Ars. Combin. 103, 239\u2013256 (2012)","journal-title":"Ars. Combin."},{"key":"1643_CR13","unstructured":"Fu, Z.-L.: The number of Latin rectangles, Math. Practice Theory\u00a0(2) (1992) 40\u201341, in Chinese"},{"key":"1643_CR14","doi-asserted-by":"crossref","unstructured":"Gessel, I.M.: Counting three-line Latin rectangles, in: Proceedings of the Colloque de Combinatoire \u00c9num\u00e9rative, Springer, (1986)","DOI":"10.1007\/BFb0072512"},{"issue":"1","key":"1643_CR15","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1090\/S0273-0979-1987-15465-6","volume":"16","author":"IM Gessel","year":"1987","unstructured":"Gessel, I.M.: Counting Latin rectangles. Bull. Amer. Math. Soc. 16(1), 79\u201383 (1987)","journal-title":"Bull. Amer. Math. Soc."},{"issue":"2","key":"1643_CR16","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1137\/090773751","volume":"24","author":"DG Glynn","year":"2010","unstructured":"Glynn, D.G.: The conjectures of Alon-Tarsi and Rota in dimension prime minus one. SIAM J. Discrete Math. 24(2), 394\u2013399 (2010)","journal-title":"SIAM J. Discrete Math."},{"key":"1643_CR17","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0095-8956(90)90128-M","volume":"48","author":"CD Godsil","year":"1990","unstructured":"Godsil, C.D., McKay, B.D.: Asymptotic enumeration of Latin rectangles. J. Combin. Theory Ser. B 48, 19\u201344 (1990)","journal-title":"J. Combin. Theory Ser. B"},{"key":"1643_CR18","volume-title":"Combinatorial enumeration","author":"IP Goulden","year":"1983","unstructured":"Goulden, I.P., Jackson, D.M.: Combinatorial enumeration. Wiley, New York (1983)"},{"issue":"2","key":"1643_CR19","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1112\/plms\/s2-31.1.329","volume":"31","author":"SM Jacob","year":"1930","unstructured":"Jacob, S.M.: The enumeration of the Latin rectangle of depth three by means of a formula of reduction, with other theorems relating to non-clashing substitutions and Latin squares. Proc. London Math. Soc. 31(2), 329\u2013354 (1930)","journal-title":"Proc. London Math. Soc."},{"key":"1643_CR20","first-page":"119","volume":"33","author":"SM Kerawala","year":"1941","unstructured":"Kerawala, S.M.: The enumeration of the Latin rectangle of depth three by means of a difference equation. Bull. Calcutta Math. Soc. 33, 119\u2013127 (1941)","journal-title":"Bull. Calcutta Math. Soc."},{"issue":"3","key":"1643_CR21","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1080\/00150517.1973.12430820","volume":"11","author":"FW Light Jr","year":"1973","unstructured":"Light Jr, F.W.: A procedure for the enumeration of $$4 \\times n$$ 4 \u00d7 n Latin rectangles. Fibonacci Quart. 11(3), 241\u2013246 (1973)","journal-title":"Fibonacci Quart."},{"key":"1643_CR22","first-page":"262","volume":"16","author":"PA MacMahon","year":"1898","unstructured":"MacMahon, P.A.: A new method in combinatory analysis, with applications to Latin squares and associated questions. Trans. Camb. Phil. Soc. 16, 262\u2013290 (1898)","journal-title":"Trans. Camb. Phil. Soc."},{"key":"1643_CR23","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s00026-005-0261-7","volume":"9","author":"BD McKay","year":"2005","unstructured":"McKay, B.D., Wanless, I.M.: On the number of Latin squares. Ann. Comb. 9, 335\u2013344 (2005)","journal-title":"Ann. Comb."},{"key":"1643_CR24","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/0012-365X(82)90129-7","volume":"40","author":"WOJ Moser","year":"1982","unstructured":"Moser, W.O.J.: A generalization of Riordan\u2019s formula for $$3 \\times n$$ 3 \u00d7 n Latin rectangles. Discrete Math. 40, 311\u2013313 (1982)","journal-title":"Discrete Math."},{"key":"1643_CR25","doi-asserted-by":"crossref","unstructured":"Pranesachar, C.R.: Enumeration of Latin rectangles via SDR\u2019s, in: Lecture Notes in Math., Springer, pp. 380\u2013390 (1981)","DOI":"10.1007\/BFb0092284"},{"issue":"8","key":"1643_CR26","doi-asserted-by":"crossref","first-page":"450","DOI":"10.2307\/2306012","volume":"51","author":"J Riordan","year":"1944","unstructured":"Riordan, J.: Three-line Latin rectangles. Amer. Math. Monthly 51(8), 450\u2013452 (1944)","journal-title":"Amer. Math. Monthly"},{"issue":"1","key":"1643_CR27","doi-asserted-by":"crossref","first-page":"18","DOI":"10.2307\/2306080","volume":"53","author":"J Riordan","year":"1946","unstructured":"Riordan, J.: Three-line Latin rectangles-II. Amer. Math. Monthly 53(1), 18\u201320 (1946)","journal-title":"Amer. Math. Monthly"},{"issue":"3","key":"1643_CR28","doi-asserted-by":"crossref","first-page":"159","DOI":"10.2307\/2308187","volume":"59","author":"J Riordan","year":"1952","unstructured":"Riordan, J.: A recurrence relation for three-line Latin rectangles. Amer. Math. Monthly 59(3), 159\u2013162 (1952)","journal-title":"Amer. Math. Monthly"},{"key":"1643_CR29","volume-title":"An introduction to combinatorial analysis","author":"J Riordan","year":"1958","unstructured":"Riordan, J.: An introduction to combinatorial analysis. Wiley, New York (1958)"},{"key":"1643_CR30","doi-asserted-by":"crossref","unstructured":"Rota, G.-C.: On the foundations of combinatorial theory. I. Theory of M\u00f6bius functions, Z. Wahrsch. und Verw. Gebiete 2:340\u2013368 (1964)","DOI":"10.1007\/BF00531932"},{"key":"1643_CR31","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/0012-365X(92)90722-R","volume":"110","author":"J-Y Shao","year":"1992","unstructured":"Shao, J.-Y., Wei, W.-D.: A formula for the number of Latin squares. Discrete Math. 110, 293\u2013296 (1992)","journal-title":"Discrete Math."},{"key":"1643_CR32","first-page":"8","volume":"2","author":"VS Shevelev","year":"1991","unstructured":"Shevelev, V.S.: The Riordan generalized formula for three-line Latin rectangles and its applications. DAN of the Ukraine 2, 8\u201312 (1991). in Russian","journal-title":"DAN of the Ukraine"},{"issue":"5","key":"1643_CR33","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1006\/eujc.1998.0221","volume":"19","author":"I Skau","year":"1998","unstructured":"Skau, I.: A note on the asymptotic number of Latin rectangles. European J. Combin. 19(5), 617\u2013620 (1998)","journal-title":"European J. Combin."},{"key":"1643_CR34","unstructured":"Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http:\/\/www.research.att.com\/~njas\/sequences\/"},{"key":"1643_CR35","unstructured":"Stones D.\u00a0S.: On the number of Latin rectangles, Ph.D. thesis, Monash University (2010). Available from http:\/\/arrow.monash.edu.au\/hdl\/1959.1\/167114"},{"key":"1643_CR36","doi-asserted-by":"crossref","first-page":"A1","DOI":"10.37236\/487","volume":"17","author":"DS Stones","year":"2010","unstructured":"Stones, D.S.: The many formulae for the number of Latin rectangles. Electron. J. Combin. 17, A1 (2010)","journal-title":"Electron. J. Combin."},{"key":"1643_CR37","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1137\/110797540","volume":"26","author":"DS Stones","year":"2012","unstructured":"Stones, D.S.: Formulae for the Alon-Tarsi Conjecture. SIAM J. Discrete Math. 26, 65\u201370 (2012)","journal-title":"SIAM J. Discrete Math."},{"key":"1643_CR38","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1002\/jcd.20309","volume":"20","author":"DS Stones","year":"2012","unstructured":"Stones, D.S., Vojt\u011bchovsk\u00fd, P., Wanless, I.M.: Cycle structure of autotopisms of quasigroups and Latin squares. J. Combin. Des. 20, 227\u2013263 (2012)","journal-title":"J. Combin. Des."},{"issue":"2","key":"1643_CR39","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.jcta.2009.03.019","volume":"117","author":"DS Stones","year":"2010","unstructured":"Stones, D.S., Wanless, I.M.: Divisors of the number of Latin rectangles. J. Combin. Theory Ser. A 117(2), 204\u2013215 (2010)","journal-title":"J. Combin. Theory Ser. A"},{"key":"1643_CR40","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s00026-012-0137-6","volume":"16","author":"DS Stones","year":"2012","unstructured":"Stones, D.S., Wanless, I.M.: A congruence connecting Latin rectangles and partial orthomorphisms. Ann. Comb. 16, 349\u2013365 (2012)","journal-title":"Ann. Comb."},{"issue":"4","key":"1643_CR41","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.ffa.2010.04.001","volume":"16","author":"DS Stones","year":"2010","unstructured":"Stones, D.S., Wanless, I.M.: Compound orthomorphisms of the cyclic group. Finite Fields Appl. 16(4), 277\u2013289 (2010)","journal-title":"Finite Fields Appl."},{"key":"1643_CR42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1215\/00277630-1543769","volume":"205","author":"DS Stones","year":"2012","unstructured":"Stones, D.S., Wanless, I.M.: How not to prove the Alon-Tarsi Conjecture. Nagoya Math. J. 205, 1\u201324 (2012)","journal-title":"Nagoya Math. J."},{"key":"1643_CR43","volume-title":"A course in combinatorics","author":"JH Lint van","year":"1992","unstructured":"van Lint, J.H., Wilson, R.M.: A course in combinatorics. Cambridge University Press, Cambridge (1992)"},{"key":"1643_CR44","unstructured":"Yamamoto, K.: Asymptotic number of Latin rectangles and the symbolic method, S\u00fbgaku 2 (1949)"},{"issue":"1","key":"1643_CR45","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/0022-247X(82)90029-4","volume":"85","author":"D Zeilberger","year":"1982","unstructured":"Zeilberger, D.: Sister Celine\u2019s technique and its generalizations. J. Math. Anal. Appl. 85(1), 114\u2013145 (1982)","journal-title":"J. Math. Anal. Appl."}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-015-1643-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00373-015-1643-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-015-1643-1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,31]],"date-time":"2025-05-31T11:52:51Z","timestamp":1748692371000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00373-015-1643-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,14]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,5]]}},"alternative-id":["1643"],"URL":"https:\/\/doi.org\/10.1007\/s00373-015-1643-1","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"type":"print","value":"0911-0119"},{"type":"electronic","value":"1435-5914"}],"subject":[],"published":{"date-parts":[[2015,11,14]]}}}