{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T02:50:38Z","timestamp":1780627838685,"version":"3.54.1"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2011,10,11]],"date-time":"2011-10-11T00:00:00Z","timestamp":1318291200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,11]]},"abstract":"<jats:p>In this paper, we study certain pairings which are defined as follows: if <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic> are finite subsets of an arbitrary group, a <jats:italic>Wakeford\u2013Fan\u2013Losonczy pairing<\/jats:italic> from <jats:italic>B<\/jats:italic> onto <jats:italic>A<\/jats:italic> is a bijection \u03c6 : <jats:italic>B<\/jats:italic> \u2192 <jats:italic>A<\/jats:italic> such that <jats:italic>b<\/jats:italic>\u03c6(<jats:italic>b<\/jats:italic>) \u2209 <jats:italic>A<\/jats:italic>, for every <jats:italic>b<\/jats:italic> \u2208 <jats:italic>B<\/jats:italic>. The number of such pairings is denoted by \u03bc(<jats:italic>B, A<\/jats:italic>).<\/jats:p><jats:p>We investigate the quantity \u03bc(<jats:italic>B, A<\/jats:italic>) for <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic>, two finite subsets of an arbitrary group satisfying 1 \u2209 <jats:italic>B<\/jats:italic>, |<jats:italic>A<\/jats:italic>| = |<jats:italic>B<\/jats:italic>|, and the fact that the order of every element of <jats:italic>B<\/jats:italic> is \u2265 |<jats:italic>B<\/jats:italic>| + 1. Extending earlier results, we show that in this case, \u03bc(<jats:italic>B, A<\/jats:italic>) is never equal to 0. Furthermore we prove an explicit lower bound on \u03bc(<jats:italic>B, A<\/jats:italic>) in terms of |<jats:italic>B<\/jats:italic>| and the cardinality of the group generated by <jats:italic>B<\/jats:italic>, which is valid unless <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic> have a special form explicitly described. In the case <jats:italic>A<\/jats:italic> = <jats:italic>B<\/jats:italic>, our bound holds unless <jats:italic>B<\/jats:italic> is a translate of a progression.<\/jats:p>","DOI":"10.1017\/s0963548311000459","type":"journal-article","created":{"date-parts":[[2011,10,11]],"date-time":"2011-10-11T03:51:41Z","timestamp":1318305101000},"page":"855-865","source":"Crossref","is-referenced-by-count":5,"title":["Counting Certain Pairings in Arbitrary Groups"],"prefix":"10.1017","volume":"20","author":[{"given":"Y. O.","family":"HAMIDOUNE","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2011,10,11]]},"reference":[{"key":"S0963548311000459_ref3","doi-asserted-by":"crossref","first-page":"149","DOI":"10.4064\/aa-9-2-149-159","article-title":"On the addition of residue classes mod p","volume":"9","author":"Erd\u0151s","year":"1964","journal-title":"Acta Arith."},{"key":"S0963548311000459_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(84)80034-7"},{"key":"S0963548311000459_ref21","first-page":"123","article-title":"Yahya ould Hamidoune, grand Mauritanien, homme singulier, math\u00e9maticien d'exception","volume":"129","author":"Plagne","year":"2011","journal-title":"Gaz. Math."},{"key":"S0963548311000459_ref1","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1007\/BF03035809","article-title":"A theorem on the addition of residue classes: Applications to the number \u0393(k) in Waring's problem","volume":"2","author":"Chowla","year":"1935","journal-title":"Proc. Indian Acad. Sci."},{"key":"S0963548311000459_ref17","doi-asserted-by":"publisher","DOI":"10.1006\/aama.1998.0587"},{"key":"S0963548311000459_ref10","unstructured":"[10] Hamidoune Y. O. (1999) On small subset product in a group. In Structure Theory of Set-Addition, Vol. 258 of Ast\u00e9risque, pp. 281\u2013308."},{"key":"S0963548311000459_ref6","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-10.37.26"},{"key":"S0963548311000459_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(56)50032-7"},{"key":"S0963548311000459_ref14","unstructured":"[14] Hamidoune Y. O. Hyper-atoms applied to the critical pair theory. Submitted. arXiv:1102.2099v1"},{"key":"S0963548311000459_ref22","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000332"},{"key":"S0963548311000459_ref24","first-page":"403","article-title":"On canonical forms","volume":"18","author":"Wakeford","year":"1918","journal-title":"Proc. London Math. Soc."},{"key":"S0963548311000459_ref19","doi-asserted-by":"crossref","first-page":"147","DOI":"10.4064\/aa-28-2-147-156","article-title":"Sums of sets of group elements","volume":"28","author":"Olson","year":"1975","journal-title":"Acta Arith."},{"key":"S0963548311000459_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(02)00496-6"},{"key":"S0963548311000459_ref8","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1996.0028"},{"key":"S0963548311000459_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/0022-314X(84)90047-7"},{"key":"S0963548311000459_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01174162"},{"key":"S0963548311000459_ref18","article-title":"Matching theory","volume":"29","author":"Lov\u00e1sz","year":"1986","journal-title":"Ann. Discrete Math."},{"key":"S0963548311000459_ref13","unstructured":"[13] Hamidoune Y. O. Hyper-atoms and the Kemperman's critical pair theory, arXiv.0708.3581."},{"key":"S0963548311000459_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2007.01.002"},{"key":"S0963548311000459_ref11","doi-asserted-by":"publisher","DOI":"10.4064\/aa96-2-1"},{"key":"S0963548311000459_ref12","unstructured":"[12] Hamidoune Y. O. (2008) On group bijections \u03c6 with \u03c6(B) = A and \u2200a \u2208 B, a\u03c6(a) \u2209 A. arXiv:0812.2522."},{"key":"S0963548311000459_ref9","doi-asserted-by":"publisher","DOI":"10.1006\/eujc.1995.0113"},{"key":"S0963548311000459_ref23","first-page":"46","article-title":"Advanced problems and solutions: Solutions, 4466","volume":"62","author":"Scherk","year":"1955","journal-title":"Amer. Math. 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