2034:
1470:
2358:
66:
In the monograph, MacMahon found so many applications of his result, he called it "a master theorem in the Theory of
Permutations." He explained the title as follows: "a Master Theorem from the masterly and rapid fashion in which it deals with various questions otherwise troublesome to solve."
889:
509:
1785:
1134:
2045:
1107:
2540:
1766:
2773:
704:
2050:
1139:
285:
2029:{\displaystyle \det(I-TA)\,=\,\det {\begin{pmatrix}1&-t_{1}&t_{1}\\t_{2}&1&-t_{2}\\-t_{3}&t_{3}&1\end{pmatrix}}\,=\,1+{\bigl (}t_{1}t_{2}+t_{1}t_{3}+t_{2}t_{3}{\bigr )}.}
692:
228:
1465:{\displaystyle {\begin{aligned}G(2n,2n,2n)&={\bigl }(x_{2}-x_{3})^{2n}(x_{3}-x_{1})^{2n}(x_{1}-x_{2})^{2n}\\&=\,\sum _{k=0}^{2n}(-1)^{k}{\binom {2n}{k}}^{3},\end{aligned}}}
944:
627:
274:
166:
It is the MacMahon Master
Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents.
2353:{\displaystyle {\begin{aligned}G(2n,2n,2n)&={\bigl }(-1)^{3n}{\bigl (}t_{1}t_{2}+t_{1}t_{3}+t_{2}t_{3}{\bigr )}^{3n}\\&=(-1)^{n}{\binom {3n}{n,n,n}},\end{aligned}}}
2363:
where the last equality follows from the fact that we need to use an equal number of times all three terms in the power. Now equating the two formulas for coefficients
1014:
971:
2385:
995:
581:
561:
541:
2659:
2609:
1481:
2843:
83:
884:{\displaystyle \sum _{(k_{1},\dots ,k_{m})}G(k_{1},\dots ,k_{m})\,t_{1}^{k_{1}}\cdots t_{m}^{k_{m}}\,=\,{\frac {1}{\det(I_{m}-TA)}},}
17:
2891:
2580:
1475:
where the last equality follows from the fact that on the right-hand side we have the product of the following coefficients:
504:{\displaystyle G(k_{1},\dots ,k_{m})\,=\,{\bigl }\,\prod _{i=1}^{m}{\bigl (}a_{i1}x_{1}+\dots +a_{im}x_{m}{\bigl )}^{k_{i}}.}
2901:
2816:
2722:
2896:
2886:
632:
120:-Dixon identities have been known for decades, except for a Krattenthaler–Schlosser extension (1999), the proper
2756:
2703:
90:
178:
2747:
897:
2596:
586:
233:
2801:
D. Foata and G.-N. Han, A new proof of the
Garoufalidis-LĂŞ-Zeilberger Quantum MacMahon Master Theorem,
75:
39:
47:
2790:
2743:
2551:
132:
extensions were developed by Foata–Han, Konvalinka–Pak, and
Etingof–Pak. Further connections to
129:
2814:
D. Foata and G.-N. Han, Specializations and extensions of the quantum MacMahon Master
Theorem,
1102:{\displaystyle A={\begin{pmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{pmatrix}}.}
106:
144:
2668:
2618:
949:
159:
2696:
2646:
2535:{\displaystyle \sum _{k=0}^{2n}(-1)^{k}{\binom {2n}{k}}^{3}=(-1)^{n}{\binom {3n}{n,n,n}}.}
8:
1002:
151:
55:
2672:
2657:(1962). "Proofs of some 'binomial' identities by means of MacMahon's 'Master Theorem'".
2622:
2803:
2707:
2684:
2634:
980:
566:
546:
517:
2688:
2638:
2576:
89:
version. In 1962, Good found a short proof of Dixon's identity from MMT. In 1969,
2764:
2733:
2692:
2676:
2642:
2626:
1772:
155:
147:
137:
2827:
P.H. Hai and M. Lorenz, Koszul algebras and the quantum MacMahon master theorem,
2717:
1761:{\displaystyle (x_{2}-x_{3})^{2n},\ \ (x_{3}-x_{1})^{2n},\ \ (x_{1}-x_{2})^{2n},}
974:
695:
125:
102:
94:
79:
70:
The result was re-derived (with attribution) a number of times, most notably by
2768:
2774:
Proceedings of the
National Academy of Sciences of the United States of America
140:
were also found by Hai–Lorentz, Hai–Kriegk–Lorenz, Konvalinka–Pak, and others.
133:
43:
2680:
2630:
2880:
2838:
113:. Since then, MMT has become a standard tool in enumerative combinatorics.
110:
86:
2729:
1776:
277:
2821:
2039:
Therefore, by the MMT, we have a new formula for the same coefficients:
2654:
2604:
71:
54:(1916). It is often used to derive binomial identities, most notably
2848:
2832:
2808:
2795:
2778:
2841:
and I. Pak, An algebraic extension of the MacMahon master theorem,
2748:
A new multidimensional matrix inverse with applications to multiple
2785:
121:
2863:
98:
2788:, Non-commutative extensions of the MacMahon Master Theorem,
124:
of MMT remained elusive. After
Garoufalidis–Lê–Zeilberger's
105:
ideas (built on Foata's thesis) and further applications to
74:
who derived it from his multilinear generalization of the
2708:
Problèmes combinatoires de commutation et réarrangements
2607:(1962). "A short proof of MacMahon's 'Master Theorem'".
894:
where the sum runs over all nonnegative integer vectors
2379:) we obtain an equivalent version of Dixon's identity:
1823:
1029:
27:
Result in enumerative combinatorics and linear algebra
2388:
2048:
1788:
1484:
1137:
1017:
983:
952:
900:
707:
635:
589:
569:
549:
520:
288:
236:
181:
2600:, vols 1 and 2, Cambridge University Press, 1915–16.
2660:Proceedings of the Cambridge Philosophical Society
2610:Proceedings of the Cambridge Philosophical Society
2534:
2352:
2028:
1760:
1464:
1101:
989:
965:
938:
883:
686:
621:
575:
555:
535:
503:
268:
222:
2523:
2491:
2454:
2436:
2337:
2305:
1443:
1425:
1000:
687:{\displaystyle T=(\delta _{ij}t_{i})_{m\times m}}
2878:
2844:Proceedings of the American Mathematical Society
1815:
1789:
847:
2720:, An Application of MacMahon's Master Theorem,
2575:. Singapore: World Scientific. pp. viii.
2259:
2185:
2156:
2095:
2018:
1945:
1245:
1184:
480:
417:
388:
334:
629:be another set of formal variables, and let
150:re-discovered the MMT in the context of his
1936:
1932:
1814:
1810:
1377:
840:
836:
788:
393:
331:
327:
1775:. On the other hand, we can compute the
143:Finally, according to J. D. Louck, the
14:
2879:
223:{\displaystyle A=(a_{ij})_{m\times m}}
97:found a new proof of MMT by combining
2570:
2653:
2603:
939:{\displaystyle (k_{1},\dots ,k_{m})}
170:
2873:, World Sci., Hackensack, NJ, 2008.
2854:P.H. Hai, B. Kriegk and M. Lorenz,
2817:Linear Algebra and its Applications
2777:103 (2006), no. 38, 13928–13931 (
2769:The Quantum MacMahon Master Theorem
2723:SIAM Journal on Applied Mathematics
622:{\displaystyle t_{1},\ldots ,t_{m}}
543:means "the coefficient of monomial
269:{\displaystyle x_{1},\ldots ,x_{m}}
24:
2871:Unitary symmetry and combinatorics
2573:Unitary symmetry and combinatorics
2495:
2440:
2309:
1429:
25:
2913:
2763:S. Garoufalidis, T. T. Q. LĂŞ and
2714:, no. 85, Springer, Berlin, 1969.
276:be formal variables. Consider a
2847:136 (2008), no. 7, 2279–2288 (
1128:) directly from the definition:
78:. MMT was also popularized by
2831:39 (2007), no. 4, 667–676. (
2820:423 (2007), no. 2–3, 445–455 (
2807:307 (2007), no. 1, 424–431 (
2564:
2479:
2469:
2423:
2413:
2293:
2283:
2171:
2161:
2083:
2056:
1807:
1792:
1743:
1716:
1713:
1671:
1650:
1623:
1620:
1578:
1557:
1530:
1527:
1485:
1412:
1402:
1355:
1328:
1316:
1289:
1277:
1250:
1172:
1145:
933:
901:
872:
850:
785:
753:
745:
713:
669:
642:
527:
521:
324:
292:
205:
188:
128:extension (2006), a number of
13:
1:
2892:Factorial and binomial topics
2740:, John Wiley, New York, 1983.
2557:
230:be a complex matrix, and let
109:, introducing the concept of
61:
2858:-homogeneous superalgebras,
2712:Lecture Notes in Mathematics
1771:which are computed from the
50:and proved in his monograph
7:
2545:
10:
2918:
2902:Theorems in linear algebra
76:Lagrange inversion theorem
2897:Theorems in combinatorics
2887:Enumerative combinatorics
2738:Combinatorial Enumeration
2681:10.1017/S030500410003632X
2631:10.1017/S0305004100036318
1112:Compute the coefficients
40:enumerative combinatorics
32:MacMahon's master theorem
2571:Louck, James D. (2008).
46:. It was discovered by
2791:Advances in Mathematics
18:MacMahon Master theorem
2829:Bull. Lond. Math. Soc.
2536:
2412:
2354:
2030:
1762:
1466:
1401:
1103:
991:
967:
940:
885:
688:
623:
577:
557:
537:
505:
414:
270:
224:
168:
107:combinatorics on words
2537:
2389:
2355:
2031:
1763:
1467:
1378:
1104:
992:
968:
966:{\displaystyle I_{m}}
941:
886:
689:
624:
578:
558:
538:
506:
394:
271:
225:
164:
160:many-particle systems
145:theoretical physicist
2794:216 (2007), no. 1. (
2760:204 (1999), 249–279.
2757:Discrete Mathematics
2597:Combinatory analysis
2386:
2046:
1786:
1482:
1135:
1015:
981:
950:
898:
705:
633:
587:
567:
547:
518:
286:
234:
179:
52:Combinatory analysis
2860:J. Noncommut. Geom.
2726:26 (1974), 431–436.
2673:1962PCPS...58..161G
2623:1962PCPS...58..160G
2153:
2135:
2117:
1712:
1688:
1619:
1595:
1526:
1502:
1242:
1224:
1206:
835:
810:
514:(Here the notation
385:
360:
152:generating function
2804:Journal of Algebra
2784:M. Konvalinka and
2746:and M. Schlosser,
2532:
2350:
2348:
2136:
2118:
2100:
2026:
1926:
1758:
1689:
1674:
1596:
1581:
1503:
1488:
1462:
1460:
1225:
1207:
1189:
1099:
1090:
1008:Consider a matrix
987:
963:
936:
881:
814:
789:
749:
684:
619:
573:
553:
533:
501:
364:
339:
266:
220:
2582:978-981-281-472-2
2521:
2452:
2335:
1670:
1667:
1577:
1574:
1441:
990:{\displaystyle m}
876:
708:
576:{\displaystyle g}
556:{\displaystyle f}
536:{\displaystyle g}
171:Precise statement
162:. Louck writes:
138:quasideterminants
116:Although various
38:) is a result in
16:(Redirected from
2909:
2744:C. Krattenthaler
2700:
2650:
2587:
2586:
2568:
2541:
2539:
2538:
2533:
2528:
2527:
2526:
2520:
2503:
2494:
2487:
2486:
2465:
2464:
2459:
2458:
2457:
2448:
2439:
2431:
2430:
2411:
2403:
2359:
2357:
2356:
2351:
2349:
2342:
2341:
2340:
2334:
2317:
2308:
2301:
2300:
2276:
2272:
2271:
2263:
2262:
2255:
2254:
2245:
2244:
2232:
2231:
2222:
2221:
2209:
2208:
2199:
2198:
2189:
2188:
2182:
2181:
2160:
2159:
2152:
2144:
2134:
2126:
2116:
2108:
2099:
2098:
2035:
2033:
2032:
2027:
2022:
2021:
2015:
2014:
2005:
2004:
1992:
1991:
1982:
1981:
1969:
1968:
1959:
1958:
1949:
1948:
1931:
1930:
1918:
1917:
1906:
1905:
1889:
1888:
1869:
1868:
1855:
1854:
1843:
1842:
1773:binomial theorem
1767:
1765:
1764:
1759:
1754:
1753:
1741:
1740:
1728:
1727:
1711:
1697:
1687:
1682:
1668:
1665:
1661:
1660:
1648:
1647:
1635:
1634:
1618:
1604:
1594:
1589:
1575:
1572:
1568:
1567:
1555:
1554:
1542:
1541:
1525:
1511:
1501:
1496:
1471:
1469:
1468:
1463:
1461:
1454:
1453:
1448:
1447:
1446:
1437:
1428:
1420:
1419:
1400:
1392:
1370:
1366:
1365:
1353:
1352:
1340:
1339:
1327:
1326:
1314:
1313:
1301:
1300:
1288:
1287:
1275:
1274:
1262:
1261:
1249:
1248:
1241:
1233:
1223:
1215:
1205:
1197:
1188:
1187:
1108:
1106:
1105:
1100:
1095:
1094:
1003:Dixon's identity
996:
994:
993:
988:
972:
970:
969:
964:
962:
961:
945:
943:
942:
937:
932:
931:
913:
912:
890:
888:
887:
882:
877:
875:
862:
861:
842:
834:
833:
832:
822:
809:
808:
807:
797:
784:
783:
765:
764:
748:
744:
743:
725:
724:
693:
691:
690:
685:
683:
682:
667:
666:
657:
656:
628:
626:
625:
620:
618:
617:
599:
598:
582:
580:
579:
574:
562:
560:
559:
554:
542:
540:
539:
534:
510:
508:
507:
502:
497:
496:
495:
494:
484:
483:
476:
475:
466:
465:
444:
443:
434:
433:
421:
420:
413:
408:
392:
391:
384:
383:
382:
372:
359:
358:
357:
347:
338:
337:
323:
322:
304:
303:
275:
273:
272:
267:
265:
264:
246:
245:
229:
227:
226:
221:
219:
218:
203:
202:
156:angular momentum
154:approach to the
148:Julian Schwinger
56:Dixon's identity
30:In mathematics,
21:
2917:
2916:
2912:
2911:
2910:
2908:
2907:
2906:
2877:
2876:
2862:2 (2008) 1–51 (
2594:P.A. MacMahon,
2591:
2590:
2583:
2569:
2565:
2560:
2548:
2522:
2504:
2496:
2490:
2489:
2488:
2482:
2478:
2460:
2453:
2441:
2435:
2434:
2433:
2432:
2426:
2422:
2404:
2393:
2387:
2384:
2383:
2347:
2346:
2336:
2318:
2310:
2304:
2303:
2302:
2296:
2292:
2274:
2273:
2264:
2258:
2257:
2256:
2250:
2246:
2240:
2236:
2227:
2223:
2217:
2213:
2204:
2200:
2194:
2190:
2184:
2183:
2174:
2170:
2155:
2154:
2145:
2140:
2127:
2122:
2109:
2104:
2094:
2093:
2086:
2049:
2047:
2044:
2043:
2017:
2016:
2010:
2006:
2000:
1996:
1987:
1983:
1977:
1973:
1964:
1960:
1954:
1950:
1944:
1943:
1925:
1924:
1919:
1913:
1909:
1907:
1901:
1897:
1891:
1890:
1884:
1880:
1875:
1870:
1864:
1860:
1857:
1856:
1850:
1846:
1844:
1838:
1834:
1829:
1819:
1818:
1787:
1784:
1783:
1746:
1742:
1736:
1732:
1723:
1719:
1698:
1693:
1683:
1678:
1653:
1649:
1643:
1639:
1630:
1626:
1605:
1600:
1590:
1585:
1560:
1556:
1550:
1546:
1537:
1533:
1512:
1507:
1497:
1492:
1483:
1480:
1479:
1459:
1458:
1449:
1442:
1430:
1424:
1423:
1422:
1421:
1415:
1411:
1393:
1382:
1368:
1367:
1358:
1354:
1348:
1344:
1335:
1331:
1319:
1315:
1309:
1305:
1296:
1292:
1280:
1276:
1270:
1266:
1257:
1253:
1244:
1243:
1234:
1229:
1216:
1211:
1198:
1193:
1183:
1182:
1175:
1138:
1136:
1133:
1132:
1089:
1088:
1083:
1075:
1069:
1068:
1063:
1058:
1049:
1048:
1040:
1035:
1025:
1024:
1016:
1013:
1012:
1006:
982:
979:
978:
975:identity matrix
957:
953:
951:
948:
947:
927:
923:
908:
904:
899:
896:
895:
857:
853:
846:
841:
828:
824:
823:
818:
803:
799:
798:
793:
779:
775:
760:
756:
739:
735:
720:
716:
712:
706:
703:
702:
696:diagonal matrix
672:
668:
662:
658:
649:
645:
634:
631:
630:
613:
609:
594:
590:
588:
585:
584:
568:
565:
564:
548:
545:
544:
519:
516:
515:
490:
486:
485:
479:
478:
477:
471:
467:
458:
454:
439:
435:
426:
422:
416:
415:
409:
398:
387:
386:
378:
374:
373:
368:
353:
349:
348:
343:
333:
332:
318:
314:
299:
295:
287:
284:
283:
260:
256:
241:
237:
235:
232:
231:
208:
204:
195:
191:
180:
177:
176:
173:
64:
28:
23:
22:
15:
12:
11:
5:
2915:
2905:
2904:
2899:
2894:
2889:
2875:
2874:
2867:
2852:
2836:
2825:
2812:
2799:
2782:
2761:
2741:
2727:
2715:
2706:and D. Foata,
2701:
2667:(1): 161–162.
2651:
2601:
2589:
2588:
2581:
2562:
2561:
2559:
2556:
2555:
2554:
2547:
2544:
2543:
2542:
2531:
2525:
2519:
2516:
2513:
2510:
2507:
2502:
2499:
2493:
2485:
2481:
2477:
2474:
2471:
2468:
2463:
2456:
2451:
2447:
2444:
2438:
2429:
2425:
2421:
2418:
2415:
2410:
2407:
2402:
2399:
2396:
2392:
2361:
2360:
2345:
2339:
2333:
2330:
2327:
2324:
2321:
2316:
2313:
2307:
2299:
2295:
2291:
2288:
2285:
2282:
2279:
2277:
2275:
2270:
2267:
2261:
2253:
2249:
2243:
2239:
2235:
2230:
2226:
2220:
2216:
2212:
2207:
2203:
2197:
2193:
2187:
2180:
2177:
2173:
2169:
2166:
2163:
2158:
2151:
2148:
2143:
2139:
2133:
2130:
2125:
2121:
2115:
2112:
2107:
2103:
2097:
2092:
2089:
2087:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2051:
2037:
2036:
2025:
2020:
2013:
2009:
2003:
1999:
1995:
1990:
1986:
1980:
1976:
1972:
1967:
1963:
1957:
1953:
1947:
1942:
1939:
1935:
1929:
1923:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1893:
1892:
1887:
1883:
1879:
1876:
1874:
1871:
1867:
1863:
1859:
1858:
1853:
1849:
1845:
1841:
1837:
1833:
1830:
1828:
1825:
1824:
1822:
1817:
1813:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1769:
1768:
1757:
1752:
1749:
1745:
1739:
1735:
1731:
1726:
1722:
1718:
1715:
1710:
1707:
1704:
1701:
1696:
1692:
1686:
1681:
1677:
1673:
1664:
1659:
1656:
1652:
1646:
1642:
1638:
1633:
1629:
1625:
1622:
1617:
1614:
1611:
1608:
1603:
1599:
1593:
1588:
1584:
1580:
1571:
1566:
1563:
1559:
1553:
1549:
1545:
1540:
1536:
1532:
1529:
1524:
1521:
1518:
1515:
1510:
1506:
1500:
1495:
1491:
1487:
1473:
1472:
1457:
1452:
1445:
1440:
1436:
1433:
1427:
1418:
1414:
1410:
1407:
1404:
1399:
1396:
1391:
1388:
1385:
1381:
1376:
1373:
1371:
1369:
1364:
1361:
1357:
1351:
1347:
1343:
1338:
1334:
1330:
1325:
1322:
1318:
1312:
1308:
1304:
1299:
1295:
1291:
1286:
1283:
1279:
1273:
1269:
1265:
1260:
1256:
1252:
1247:
1240:
1237:
1232:
1228:
1222:
1219:
1214:
1210:
1204:
1201:
1196:
1192:
1186:
1181:
1178:
1176:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1140:
1110:
1109:
1098:
1093:
1087:
1084:
1082:
1079:
1076:
1074:
1071:
1070:
1067:
1064:
1062:
1059:
1057:
1054:
1051:
1050:
1047:
1044:
1041:
1039:
1036:
1034:
1031:
1030:
1028:
1023:
1020:
1005:
1001:Derivation of
999:
986:
960:
956:
935:
930:
926:
922:
919:
916:
911:
907:
903:
892:
891:
880:
874:
871:
868:
865:
860:
856:
852:
849:
845:
839:
831:
827:
821:
817:
813:
806:
802:
796:
792:
787:
782:
778:
774:
771:
768:
763:
759:
755:
752:
747:
742:
738:
734:
731:
728:
723:
719:
715:
711:
681:
678:
675:
671:
665:
661:
655:
652:
648:
644:
641:
638:
616:
612:
608:
605:
602:
597:
593:
572:
552:
532:
529:
526:
523:
512:
511:
500:
493:
489:
482:
474:
470:
464:
461:
457:
453:
450:
447:
442:
438:
432:
429:
425:
419:
412:
407:
404:
401:
397:
390:
381:
377:
371:
367:
363:
356:
352:
346:
342:
336:
330:
326:
321:
317:
313:
310:
307:
302:
298:
294:
291:
263:
259:
255:
252:
249:
244:
240:
217:
214:
211:
207:
201:
198:
194:
190:
187:
184:
172:
169:
134:Koszul algebra
130:noncommutative
63:
60:
48:Percy MacMahon
44:linear algebra
26:
9:
6:
4:
3:
2:
2914:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2884:
2882:
2872:
2868:
2865:
2861:
2857:
2853:
2850:
2846:
2845:
2840:
2837:
2834:
2830:
2826:
2823:
2819:
2818:
2813:
2810:
2806:
2805:
2800:
2797:
2793:
2792:
2787:
2783:
2780:
2776:
2775:
2770:
2766:
2765:D. Zeilberger
2762:
2759:
2758:
2753:
2751:
2745:
2742:
2739:
2735:
2734:D. M. Jackson
2731:
2728:
2725:
2724:
2719:
2716:
2713:
2709:
2705:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2661:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2611:
2606:
2602:
2599:
2598:
2593:
2592:
2584:
2578:
2574:
2567:
2563:
2553:
2550:
2549:
2529:
2517:
2514:
2511:
2508:
2505:
2500:
2497:
2483:
2475:
2472:
2466:
2461:
2449:
2445:
2442:
2427:
2419:
2416:
2408:
2405:
2400:
2397:
2394:
2390:
2382:
2381:
2380:
2378:
2374:
2370:
2366:
2343:
2331:
2328:
2325:
2322:
2319:
2314:
2311:
2297:
2289:
2286:
2280:
2278:
2268:
2265:
2251:
2247:
2241:
2237:
2233:
2228:
2224:
2218:
2214:
2210:
2205:
2201:
2195:
2191:
2178:
2175:
2167:
2164:
2149:
2146:
2141:
2137:
2131:
2128:
2123:
2119:
2113:
2110:
2105:
2101:
2090:
2088:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2053:
2042:
2041:
2040:
2023:
2011:
2007:
2001:
1997:
1993:
1988:
1984:
1978:
1974:
1970:
1965:
1961:
1955:
1951:
1940:
1937:
1933:
1927:
1921:
1914:
1910:
1902:
1898:
1894:
1885:
1881:
1877:
1872:
1865:
1861:
1851:
1847:
1839:
1835:
1831:
1826:
1820:
1811:
1804:
1801:
1798:
1795:
1782:
1781:
1780:
1778:
1774:
1755:
1750:
1747:
1737:
1733:
1729:
1724:
1720:
1708:
1705:
1702:
1699:
1694:
1690:
1684:
1679:
1675:
1662:
1657:
1654:
1644:
1640:
1636:
1631:
1627:
1615:
1612:
1609:
1606:
1601:
1597:
1591:
1586:
1582:
1569:
1564:
1561:
1551:
1547:
1543:
1538:
1534:
1522:
1519:
1516:
1513:
1508:
1504:
1498:
1493:
1489:
1478:
1477:
1476:
1455:
1450:
1438:
1434:
1431:
1416:
1408:
1405:
1397:
1394:
1389:
1386:
1383:
1379:
1374:
1372:
1362:
1359:
1349:
1345:
1341:
1336:
1332:
1323:
1320:
1310:
1306:
1302:
1297:
1293:
1284:
1281:
1271:
1267:
1263:
1258:
1254:
1238:
1235:
1230:
1226:
1220:
1217:
1212:
1208:
1202:
1199:
1194:
1190:
1179:
1177:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1142:
1131:
1130:
1129:
1127:
1123:
1119:
1115:
1096:
1091:
1085:
1080:
1077:
1072:
1065:
1060:
1055:
1052:
1045:
1042:
1037:
1032:
1026:
1021:
1018:
1011:
1010:
1009:
1004:
998:
984:
976:
958:
954:
928:
924:
920:
917:
914:
909:
905:
878:
869:
866:
863:
858:
854:
843:
837:
829:
825:
819:
815:
811:
804:
800:
794:
790:
780:
776:
772:
769:
766:
761:
757:
750:
740:
736:
732:
729:
726:
721:
717:
709:
701:
700:
699:
697:
679:
676:
673:
663:
659:
653:
650:
646:
639:
636:
614:
610:
606:
603:
600:
595:
591:
570:
550:
530:
524:
498:
491:
487:
472:
468:
462:
459:
455:
451:
448:
445:
440:
436:
430:
427:
423:
410:
405:
402:
399:
395:
379:
375:
369:
365:
361:
354:
350:
344:
340:
328:
319:
315:
311:
308:
305:
300:
296:
289:
282:
281:
280:
279:
261:
257:
253:
250:
247:
242:
238:
215:
212:
209:
199:
196:
192:
185:
182:
167:
163:
161:
157:
153:
149:
146:
141:
139:
135:
131:
127:
123:
119:
114:
112:
108:
104:
100:
96:
92:
88:
85:
82:who found an
81:
77:
73:
68:
59:
57:
53:
49:
45:
41:
37:
33:
19:
2870:
2869:J.D. Louck,
2859:
2855:
2842:
2828:
2815:
2802:
2789:
2772:
2755:
2749:
2737:
2730:I.P. Goulden
2721:
2711:
2664:
2658:
2614:
2608:
2595:
2572:
2566:
2376:
2372:
2368:
2364:
2362:
2038:
1779:explicitly:
1770:
1474:
1125:
1121:
1117:
1113:
1111:
1007:
973:denotes the
893:
513:
174:
165:
142:
117:
115:
87:power series
69:
65:
51:
35:
31:
29:
1777:determinant
278:coefficient
84:exponential
2881:Categories
2839:P. Etingof
2718:L. Carlitz
2704:P. Cartier
2697:0108.25105
2655:Good, I.J.
2647:0108.25104
2617:(1): 160.
2605:Good, I.J.
2558:References
158:theory of
72:I. J. Good
62:Background
2689:122896760
2639:124876088
2552:Permanent
2473:−
2417:−
2391:∑
2287:−
2165:−
1895:−
1878:−
1832:−
1799:−
1730:−
1706:−
1637:−
1613:−
1544:−
1520:−
1406:−
1380:∑
1342:−
1303:−
1264:−
1078:−
1053:−
1043:−
918:…
864:−
812:⋯
770:…
730:…
710:∑
677:×
647:δ
604:…
583:".) Let
449:⋯
396:∏
362:⋯
309:…
251:…
213:×
103:bijective
99:algebraic
2546:See also
2375:, 2
2371:, 2
1124:, 2
1120:, 2
977:of size
698:. Then
122:q-analog
2752:-series
2669:Bibcode
2619:Bibcode
126:quantum
91:Cartier
80:Carlitz
2864:eprint
2849:eprint
2833:eprint
2822:eprint
2809:eprint
2796:eprint
2786:I. Pak
2779:eprint
2695:
2687:
2645:
2637:
2579:
1669:
1666:
1576:
1573:
946:, and
111:traces
2685:S2CID
2635:S2CID
694:be a
95:Foata
2732:and
2577:ISBN
175:Let
136:and
101:and
93:and
42:and
2693:Zbl
2677:doi
2643:Zbl
2627:doi
1816:det
1790:det
848:det
563:in
36:MMT
2883::
2866:).
2851:).
2835:).
2824:).
2811:).
2798:).
2781:).
2771:,
2767:,
2754:,
2736:,
2710:,
2691:.
2683:.
2675:.
2665:58
2663:.
2641:.
2633:.
2625:.
2615:58
2613:.
2367:(2
1116:(2
997:.
58:.
2856:N
2750:q
2699:.
2679::
2671::
2649:.
2629::
2621::
2585:.
2530:.
2524:)
2518:n
2515:,
2512:n
2509:,
2506:n
2501:n
2498:3
2492:(
2484:n
2480:)
2476:1
2470:(
2467:=
2462:3
2455:)
2450:k
2446:n
2443:2
2437:(
2428:k
2424:)
2420:1
2414:(
2409:n
2406:2
2401:0
2398:=
2395:k
2377:n
2373:n
2369:n
2365:G
2344:,
2338:)
2332:n
2329:,
2326:n
2323:,
2320:n
2315:n
2312:3
2306:(
2298:n
2294:)
2290:1
2284:(
2281:=
2269:n
2266:3
2260:)
2252:3
2248:t
2242:2
2238:t
2234:+
2229:3
2225:t
2219:1
2215:t
2211:+
2206:2
2202:t
2196:1
2192:t
2186:(
2179:n
2176:3
2172:)
2168:1
2162:(
2157:]
2150:n
2147:2
2142:3
2138:t
2132:n
2129:2
2124:2
2120:t
2114:n
2111:2
2106:1
2102:t
2096:[
2091:=
2084:)
2081:n
2078:2
2075:,
2072:n
2069:2
2066:,
2063:n
2060:2
2057:(
2054:G
2024:.
2019:)
2012:3
2008:t
2002:2
1998:t
1994:+
1989:3
1985:t
1979:1
1975:t
1971:+
1966:2
1962:t
1956:1
1952:t
1946:(
1941:+
1938:1
1934:=
1928:)
1922:1
1915:3
1911:t
1903:3
1899:t
1886:2
1882:t
1873:1
1866:2
1862:t
1852:1
1848:t
1840:1
1836:t
1827:1
1821:(
1812:=
1808:)
1805:A
1802:T
1796:I
1793:(
1756:,
1751:n
1748:2
1744:)
1738:2
1734:x
1725:1
1721:x
1717:(
1714:]
1709:k
1703:n
1700:2
1695:2
1691:x
1685:k
1680:1
1676:x
1672:[
1663:,
1658:n
1655:2
1651:)
1645:1
1641:x
1632:3
1628:x
1624:(
1621:]
1616:k
1610:n
1607:2
1602:1
1598:x
1592:k
1587:3
1583:x
1579:[
1570:,
1565:n
1562:2
1558:)
1552:3
1548:x
1539:2
1535:x
1531:(
1528:]
1523:k
1517:n
1514:2
1509:3
1505:x
1499:k
1494:2
1490:x
1486:[
1456:,
1451:3
1444:)
1439:k
1435:n
1432:2
1426:(
1417:k
1413:)
1409:1
1403:(
1398:n
1395:2
1390:0
1387:=
1384:k
1375:=
1363:n
1360:2
1356:)
1350:2
1346:x
1337:1
1333:x
1329:(
1324:n
1321:2
1317:)
1311:1
1307:x
1298:3
1294:x
1290:(
1285:n
1282:2
1278:)
1272:3
1268:x
1259:2
1255:x
1251:(
1246:]
1239:n
1236:2
1231:3
1227:x
1221:n
1218:2
1213:2
1209:x
1203:n
1200:2
1195:1
1191:x
1185:[
1180:=
1173:)
1170:n
1167:2
1164:,
1161:n
1158:2
1155:,
1152:n
1149:2
1146:(
1143:G
1126:n
1122:n
1118:n
1114:G
1097:.
1092:)
1086:0
1081:1
1073:1
1066:1
1061:0
1056:1
1046:1
1038:1
1033:0
1027:(
1022:=
1019:A
985:m
959:m
955:I
934:)
929:m
925:k
921:,
915:,
910:1
906:k
902:(
879:,
873:)
870:A
867:T
859:m
855:I
851:(
844:1
838:=
830:m
826:k
820:m
816:t
805:1
801:k
795:1
791:t
786:)
781:m
777:k
773:,
767:,
762:1
758:k
754:(
751:G
746:)
741:m
737:k
733:,
727:,
722:1
718:k
714:(
680:m
674:m
670:)
664:i
660:t
654:j
651:i
643:(
640:=
637:T
615:m
611:t
607:,
601:,
596:1
592:t
571:g
551:f
531:g
528:]
525:f
522:[
499:.
492:i
488:k
481:)
473:m
469:x
463:m
460:i
456:a
452:+
446:+
441:1
437:x
431:1
428:i
424:a
418:(
411:m
406:1
403:=
400:i
389:]
380:m
376:k
370:m
366:x
355:1
351:k
345:1
341:x
335:[
329:=
325:)
320:m
316:k
312:,
306:,
301:1
297:k
293:(
290:G
262:m
258:x
254:,
248:,
243:1
239:x
216:m
210:m
206:)
200:j
197:i
193:a
189:(
186:=
183:A
118:q
34:(
20:)
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