995:
graph is formed by removing the perfect matching formed by the male-female couples from a complete bipartite graph that connects every man to every woman. Any valid seating arrangement can be described by the sequence of people in order around the table, which forms a
Hamiltonian cycle in the graph. However, two Hamiltonian cycles are considered to be equivalent if they connect the same vertices in the same cyclic order regardless of the starting vertex, while in the ménage problem the starting position is considered significant: if, as in
660:, solutions to the ménage problem took the form of first finding all seating arrangements for the women and then counting, for each of these partial seating arrangements, the number of ways of completing it by seating the men away from their partners. Bogart and Doyle argued that Touchard's formula may be derived directly by considering all seating arrangements at once rather than by factoring out the participation of the women. However,
31:
954:
1029:
A second graph-theoretic description of the problem is also possible. Once the women have been seated, the possible seating arrangements for the remaining men can be described as perfect matchings in a graph formed by removing a single
Hamiltonian cycle from a complete bipartite graph; the graph has
1113:
However, the knot listing problem has some additional symmetries not present in the ménage problem: one obtains different Dowker notations for the same knot diagram if one begins the labeling at a different crossing point, and these different notations should all be counted as representing the same
994:
vertices of two colors, and each vertex of one color is connected to all but one of the vertices of the other color. In the case of the ménage problem, the vertices of the graph represent men and women, and the edges represent pairs of men and women who are allowed to sit next to each other. This
1090:. In a reduced diagram, the two labels at a crossing cannot be consecutive, so the set of pairs of labels at each crossing, used in Dowker notation to represent the knot, can be interpreted as a perfect matching in a graph that has a vertex for every number in the range from 1 to 2
999:'s tea party, all the guests shift their positions by one seat, it is considered a different seating arrangement even though it is described by the same cycle. Therefore, the number of oriented Hamiltonian cycles in a crown graph is smaller by a factor of 2
1110:, this matching is enough to describe the knot diagram itself; for other knots, an additional positive or negative sign needs to be specified for each crossing pair to determine which of the two strands of the crossing lies above the other strand.
1106:. This graph is formed by removing a Hamiltonian cycle (connecting consecutive numbers) from a complete bipartite graph (connecting all pairs of numbers with different parity), and so it has a number of matchings equal to a ménage number. For
300:
497:
805:
940:
664:
found the even more straightforward ladies-first solution described above by making use of a few of Bogart and Doyle's ideas (although they took care to recast the argument in non-gendered language).
50:
asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so that men and women alternate and nobody sits next to his or her partner. (
1030:
edges connecting open seats to men, and the removal of the cycle corresponds to forbidding the men to sit in either of the open seats adjacent to their wives. The problem of counting matchings in a
606:
34:
A table with ten place settings. There are 3120 different ways in which five male-female couples can sit at this table such that men and women alternate and nobody sits next to their partner.
957:
Crown graphs with six, eight, and ten vertices. The outer cycle of each graph forms a
Hamiltonian cycle; the eight and ten vertex graphs also have other Hamiltonian cycles.
143:
1878:
1578:
355:
676:
1688:
1133:
1022:
521:
81:
816:
1386:
996:
650:
1497:
1898:
1840:
1513:
1398:
1346:
1994:
1278:
1067:
1803:
1613:
535:
109:
1767:
1735:
1007: − 1)! than the ménage numbers. The sequence of numbers of cycles in these graphs (as before, starting at
653:
to arrangements in which the people seated at the endpoints of each edge of a matching are required to be a couple.
1989:
1152:
981:
1039:
615:
101:
349:! ways of seating them at a particular set of seats. For each seating arrangement for the women, there are
345:! ways of seating the women: there are two sets of seats that can be arranged for the women, and there are
1789:
1984:
635:
326:
1273:
58:
word for "household", referring here to a male-female couple.) This problem was formulated in 1891 by
1999:
1701:
1095:
1047:
1344:
Canfield, E. Rodney; Wormald, Nicholas C. (1987), "Ménage numbers, bijections and P-recursiveness",
1794:
1618:
1472:
1146:
668:
89:
96:
of numbers. Along with their applications to etiquette and knot theory, these numbers also have a
1684:"Über die Permanente gewisser zirkulanter Matrizen und damit zusammenhängender Toeplitz-Matrizen"
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1423:
1295:
1115:
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Tait's motivation for studying the ménage problem came from trying to find a complete listing of
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63:
1960:
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points where a knot crosses itself, in consecutive order along the knot, are labeled with the 2
295:{\displaystyle M_{n}=2\cdot n!\sum _{k=0}^{n}(-1)^{k}{\frac {2n}{2n-k}}{2n-k \choose k}(n-k)!.}
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solved this modified enumeration problem, showing that the number of different matchings is
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977:
59:
1941:
1921:
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1827:
1631:
1601:
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Henderson, J. R. (1975), "Permanents of (0,1)-matrices having at most two zeros per line",
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506:! factor from Touchard's formula. The resulting smaller numbers (again, starting from
70:. For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is
1978:
1760:
1674:
1557:
1467:
1427:
1360:
492:{\displaystyle A_{n}=\sum _{k=0}^{n}(-1)^{k}{\frac {2n}{2n-k}}{2n-k \choose k}(n-k)!}
39:
1705:
1592:
1544:
Holst, Lars (1991), "On the 'problème des ménages' from a probabilistic viewpoint",
17:
1493:
962:
97:
1819:
800:{\displaystyle A_{n}=nA_{n-1}+{\frac {n}{n-2}}A_{n-2}+{\frac {4(-1)^{n-1}}{n-2}}}
1435:
1149:, a different mathematical problem involving the arrangement of diners at tables
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625:
67:
1755:
1043:
1328:
1965:
1946:
1893:
1912:
1527:
1202:. More complicated recurrences had been described previously by Cayley and
935:{\displaystyle \displaystyle A_{n}=nA_{n-1}+2A_{n-2}-(n-4)A_{n-3}-A_{n-4},}
953:
1317:
International
Journal of Mathematical Education in Science and Technology
1114:
diagram. For this reason, two matchings that differ from each other by a
93:
1666:
614:
non-overlapping pairs of adjacent seats or, equivalently, the number of
309:
for this formula and into various generalized versions of the problem.
1411:
1299:
1446:(1983), "Some remarks on the permanents of circulant (0,1) matrices",
1054:
in which all but two adjacent elements of the generating row equal 1.
74:
12, 96, 3120, 115200, 5836320, 382072320, 31488549120, ... (sequence
1739:
1291:
1003:
than the number of seating arrangements, but larger by a factor of (
1657:
945:
from which the ménage numbers themselves can easily be calculated.
1377:
Dörrie, Heinrich (1965), "Lucas' Problem of the
Married Couples",
1315:
Bong, Nguyen-Huu (1998), "Lucas numbers and the menage problem",
1078:
for knot diagrams, an early form of which was used by Tait, the 2
1038:
the problem of computing ménage numbers, can be solved using the
1015:
2, 12, 312, 9600, 416880, 23879520, 1749363840, ... (sequence
1094:
and an edge between every pair of numbers that has different
1128:
1017:
516:
76:
1126:
1, 2, 5, 20, 87, 616, 4843, 44128, 444621, ... (sequence
502:
ways of seating the men; this formula simply omits the 2×
1770:(1882), "Additional note on a problem of arrangement",
1118:
should be treated as equivalent and counted only once.
1396:
Dutka, Jacques (1986), "On the problème des ménages",
961:
Solutions to the ménage problem may be interpreted in
514:
1, 2, 13, 80, 579, 4738, 43387, 439792, ... (sequence
820:
819:
679:
661:
538:
358:
146:
1639:
Kirousis, L.; Kontogeorgiou, G. (2018), "102.18 The
1470:(1956), "Knots and classes of ménage permutations",
1247:
1155:, a similar problem involving partial derangements
934:
799:
601:{\displaystyle {\frac {2n}{2n-k}}{2n-k \choose k}}
600:
491:
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568:
465:
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241:
1576:(1943), "Solution of the problème des ménages",
1219:
305:Much subsequent work has gone into alternative
27:Assignment problem in combinatorial mathematics
129:denote the number of seating arrangements for
1772:Proceedings of the Royal Society of Edinburgh
1759:. Includes (pp. 388–391) an addition by
1744:Proceedings of the Royal Society of Edinburgh
1710:Bulletin de la Société Mathématique de France
1579:Bulletin of the American Mathematical Society
1050:arising from this view of the problem is the
8:
1806:(1952), "The arithmetic of ménage numbers",
1786:An elementary solution to the ménage problem
1740:"On Professor Tait's problem of arrangement"
1379:100 Great Problems of Elementary Mathematics
1272:Bogart, Kenneth P.; Doyle, Peter G. (1986),
1729:, Paris: Gauthier-Villars, pp. 491–495
1274:"Non-sexist solution of the ménage problem"
657:
88:Mathematicians have developed formulas and
62:and independently, a few years earlier, by
330:
100:interpretation: they count the numbers of
1911:
1838:(1981), "On the "problème des ménages"",
1793:
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1359:
1255:
1046:. In the case of the ménage problem, the
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337:Ménage numbers and ladies-first solutions
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976:. A crown graph is formed by removing a
952:
649:is the immediate result of applying the
134:
92:for computing these numbers and related
29:
1896:(1958), "On the problème des ménages",
1251:
1199:
1164:
1119:
1689:Séminaire Lotharingien de Combinatoire
1183:
1171:
810:and the simpler four-term recurrence
7:
1706:"Sur deux problèmes de permutations"
1616:(1946), "The problème des ménages",
1243:
1215:
1203:
1195:
1248:Eades, Praeger & Seberry (1983)
662:Kirousis & Kontogeorgiou (2018)
327:Chebyshev polynomials of first kind
1546:Statistics and Probability Letters
572:
445:
245:
25:
1874:"Sur un problème de permutations"
949:Graph-theoretical interpretations
608:is the number of ways of forming
1682:Kräuter, Arnold Richard (1984),
651:principle of inclusion–exclusion
1899:Canadian Journal of Mathematics
1593:10.1090/S0002-9904-1943-08035-4
667:The ménage numbers satisfy the
1961:"Laisant's Recurrence Formula"
1514:Canadian Mathematical Bulletin
1399:The Mathematical Intelligencer
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1854:10.1016/S0012-365X(81)80024-6
1820:10.1215/S0012-7094-52-01904-2
1279:American Mathematical Monthly
1220:Canfield & Wormald (1987)
1784:Passmore, Amanda F. (2005),
1558:10.1016/0167-7152(91)90147-J
1392:. Translated by David Antin.
1361:10.1016/0012-365X(87)90002-1
2016:
1498:"Math + Sexism: A Problem"
1942:"Married Couples Problem"
1808:Duke Mathematical Journal
1756:10.1017/S0370164600032557
1381:, Dover, pp. 27–33,
658:Bogart & Doyle (1986)
40:combinatorial mathematics
1645:The Mathematical Gazette
1329:10.1080/0020739980290502
1098:and are non-consecutive
982:complete bipartite graph
331:Wyman & Moser (1958)
1153:Problème des rencontres
115:
108:in certain families of
1913:10.4153/cjm-1958-045-6
1879:C. R. Acad. Sci. Paris
1528:10.4153/CMB-1975-064-6
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1708:, Vie de la société,
1702:Laisant, Charles-Ange
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638:. The expression for
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1995:Recurrence relations
1841:Discrete Mathematics
1641:problème des ménages
1496:(October 28, 1986),
1448:Utilitas Mathematica
1444:Seberry, Jennifer R.
1347:Discrete Mathematics
817:
677:
536:
356:
144:
137:derived the formula
90:recurrence equations
48:problème des ménages
18:Problème des ménages
1727:Théorie des Nombres
1667:10.1017/mag.2018.27
1619:Scripta Mathematica
1473:Scripta Mathematica
1147:Oberwolfach problem
1086:numbers from 1 to 2
1068:number of crossings
1011: = 3) is
669:recurrence relation
66:in connection with
1958:Weisstein, Eric W.
1939:Weisstein, Eric W.
1440:Praeger, Cheryl E.
1412:10.1007/BF03025785
1116:cyclic permutation
1064:mathematical knots
970:Hamiltonian cycles
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656:Until the work of
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116:Touchard's formula
106:Hamiltonian cycles
64:Peter Guthrie Tait
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1990:Integer sequences
1610:Kaplansky, Irving
1574:Kaplansky, Irving
1388:978-0-486-61348-2
1108:alternating knots
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1342:
1338:
1334:
1330:
1326:
1322:
1318:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1280:
1275:
1270:
1269:
1264:
1257:
1253:
1249:
1245:
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1237:
1233:
1228:
1225:
1221:
1217:
1212:
1209:
1205:
1201:
1197:
1192:
1189:
1185:
1184:Gleick (1986)
1180:
1177:
1173:
1168:
1165:
1159:
1154:
1151:
1148:
1145:
1144:
1140:
1135:
1130:
1125:
1124:
1123:
1121:
1117:
1111:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1066:with a given
1065:
1057:
1055:
1053:
1049:
1045:
1041:
1037:
1033:
1024:
1019:
1014:
1013:
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1002:
998:
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989:
983:
979:
975:
971:
968:
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920:
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909:
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867:
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860:
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834:
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826:
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813:
812:
811:
791:
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637:
633:
627:
622:
617:
612:
587:
583:
580:
577:
574:
559:
556:
553:
550:
545:
542:
532:. The factor
531:
523:
518:
513:
512:
511:
509:
505:
486:
480:
477:
474:
460:
456:
453:
450:
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432:
429:
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423:
418:
415:
407:
399:
396:
388:
383:
380:
377:
373:
369:
364:
360:
352:
351:
350:
348:
344:
336:
334:
332:
329:was given by
328:
323:
319:
315:
310:
308:
289:
286:
280:
277:
274:
260:
256:
253:
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247:
232:
229:
226:
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218:
215:
207:
199:
196:
188:
183:
180:
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173:
169:
166:
163:
160:
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140:
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136:
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127:
123:
113:
111:
107:
103:
99:
95:
91:
83:
78:
73:
72:
71:
69:
65:
61:
60:Édouard Lucas
57:
53:
49:
45:
41:
32:
19:
1985:Permutations
1964:
1945:
1903:
1897:
1892:Wyman, Max;
1883:
1877:
1870:Touchard, J.
1845:
1839:
1814:(1): 27–30,
1811:
1807:
1785:
1775:
1771:
1768:Muir, Thomas
1747:
1743:
1736:Muir, Thomas
1726:
1713:
1709:
1693:
1687:
1648:
1644:
1643:revisited",
1640:
1623:
1617:
1583:
1577:
1549:
1545:
1518:
1512:
1501:
1477:
1471:
1451:
1447:
1436:Eades, Peter
1406:(3): 18–33,
1403:
1397:
1378:
1351:
1345:
1320:
1316:
1283:
1277:
1239:
1227:
1211:
1191:
1179:
1172:Dutka (1986)
1167:
1112:
1103:
1091:
1087:
1083:
1079:
1071:
1061:
1044:0-1 matrices
1035:
1028:
1008:
1004:
1000:
991:
984:
974:crown graphs
960:
944:
809:
666:
655:
644:
640:
631:
620:
610:
529:
527:
507:
503:
501:
346:
342:
341:There are 2×
340:
321:
317:
316:formula for
312:A different
311:
304:
130:
125:
121:
119:
87:
51:
47:
43:
37:
2000:Knot theory
1750:: 382–391,
1626:: 113–124,
1614:Riordan, J.
1480:: 228–233,
1454:: 145–159,
1244:Muir (1878)
1216:Muir (1882)
1204:Muir (1878)
1196:Muir (1882)
1058:Knot theory
1042:of certain
626:cycle graph
624:edges in a
68:knot theory
1979:Categories
1894:Moser, Leo
1658:1607.04115
1265:References
1040:permanents
1036:a fortiori
990:; it has 2
965:terms, as
325:involving
1966:MathWorld
1947:MathWorld
1886:(631–633)
1790:CiteSeerX
1778:: 187–190
1716:: 105–108
1675:126036427
1428:116433056
921:−
910:−
902:−
885:−
876:−
868:−
846:−
789:−
776:−
762:−
742:−
725:−
705:−
616:matchings
581:−
557:−
478:−
454:−
430:−
397:−
374:∑
278:−
254:−
230:−
197:−
174:∑
164:⋅
133:couples.
102:matchings
94:sequences
1872:(1934),
1738:(1878),
1725:(1891),
1704:(1891),
1141:See also
967:directed
636:vertices
1922:0095127
1862:0675360
1828:0045680
1632:0019074
1602:0009006
1566:1097978
1537:0399127
1486:0090568
1460:0703136
1420:0846991
1370:0885491
1337:1649926
1308:0856291
1300:2323022
1132:in the
1129:A002484
1021:in the
1018:A094047
980:from a
520:in the
517:A000179
80:in the
77:A059375
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1298:
1100:modulo
1096:parity
1070:, say
1048:matrix
314:umbral
307:proofs
110:graphs
56:French
52:Ménage
42:, the
1671:S2CID
1653:arXiv
1424:S2CID
1296:JSTOR
1160:Notes
1074:. In
997:Alice
1694:B11b
1383:ISBN
1134:OEIS
1023:OEIS
522:OEIS
120:Let
104:and
82:OEIS
1908:doi
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1850:doi
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1752:doi
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987:n,n
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628:of
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1364:,
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871:2
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832:=
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756:4
750:+
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588:k
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170:!
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