868:
2176:
684:
455:
1955:
1746:
695:
1584:
1142:
529:
700:
534:
1859:
1012:
1441:
1950:
1221:
928:
253:
313:
941:
The plethystic exponential can be used to provide innumerous product-sum identities. This is a consequence of a product formula for plethystic exponentials themselves. If
490:
521:
344:
1895:
1372:
1345:
1314:
1287:
1248:
1039:
900:
352:
147:
187:
167:
1044:
1590:
863:{\displaystyle {\begin{aligned}{\text{PE}}&={\frac {1}{1-x^{n}}},n\in \mathbb {N} \\{\text{PE}}\left&=1+\sum _{n\geq 1}p(n)x^{n}\end{aligned}}}
1251:
106:, the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariant of its symmetric products.
49:
2171:{\displaystyle {\text{PE}}=\prod _{k=0}^{d}\left(1-t^{k}x\right)^{(-1)^{k+1}b_{k}(X)}=\sum _{n\geq 0}P_{\operatorname {Sym} ^{n}(X)}(-t)\,x^{n}}
1453:
2241:
1774:
679:{\displaystyle {\begin{aligned}{\text{PE}}&=1\\{\text{PE}}&={\text{PE}}{\text{PE}}\\{\text{PE}}&={\text{PE}}^{-1}\end{aligned}}}
2414:
1317:
45:
1144:
The analogous product expression also holds in the many variables case. One particularly interesting case is its relation to
53:
2299:
944:
2484:
92:
1377:
2190:
1916:
88:
1167:
84:
1769:
1444:
25:
2258:
36:, translates addition into multiplication. This exponential operator appears naturally in the theory of
2189:, proposed a programme for systematically counting single and multi-trace gauge invariant operators of
905:
195:
2436:
2311:
33:
261:
72:
41:
463:
2460:
2426:
2395:
2369:
2335:
2185:
In a series of articles, a group of theoretical physicists, including Bo Feng, Amihay Hanany and
495:
318:
37:
2452:
2387:
2327:
2280:
2237:
1864:
1145:
56:
homogeneous symmetric polynomials in many variables. Its name comes from the operation called
2444:
2379:
2319:
2270:
2229:
1350:
1323:
1292:
1265:
1226:
1017:
450:{\displaystyle {\text{PE}}(x)=\exp \left(\sum _{k=1}^{\infty }{\frac {f(x^{k})}{k}}\right)}
2354:
1153:
876:
2198:
933:
The plethystic exponential can be also defined for power series rings in many variables.
2448:
2440:
2315:
117:
2202:
172:
152:
2275:
2478:
2399:
2339:
2223:
68:
2464:
1902:
29:
1741:{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}e_{n}\,t^{n}={\text{PE}}={\text{PE}}}
2355:"Plethystic Exponential Calculus and Characteristic Polynomials of Permutations"
2186:
1149:
61:
17:
492:
is the usual exponential function. It is readily verified that (writing simply
2323:
2233:
1761:
80:
2456:
2391:
2331:
2284:
2201:
singularities, this count is codified in the plethystic exponential of the
2383:
1579:{\displaystyle \sum _{n=0}^{\infty }h_{n}\,t^{n}={\text{PE}}={\text{PE}}}
1255:
103:
99:
57:
258:
the ideal consisting of power series without constant term. Then, given
2431:
2194:
1137:{\displaystyle {\text{PE}}(x)=\prod _{k=1}^{\infty }(1-x^{k})^{-a_{k}}}
76:
1447:, that can succinctly be written, using plethystic exponentials, as:
2374:
2225:
Combinatorial
Enumeration of Groups, Graphs, and Chemical Compounds
2304:
Mathematical
Proceedings of the Cambridge Philosophical Society
2259:"The number of linear, directed, rooted, and connected graphs"
1854:{\displaystyle P_{X}(t)=\sum _{k=0}^{d}b_{k}(X)\,t^{k}}
2413:
Feng, Bo; Hanany, Amihay; He, Yang-Hui (2007-03-20).
1958:
1919:
1867:
1777:
1593:
1456:
1380:
1353:
1326:
1295:
1268:
1229:
1170:
1047:
1020:
1014:
denotes a formal power series with real coefficients
947:
908:
879:
698:
532:
498:
466:
355:
321:
264:
198:
175:
155:
120:
1007:{\displaystyle f(x)=\sum _{k=1}^{\infty }a_{k}x^{k}}
2415:"Counting gauge invariants: the plethystic program"
2170:
1944:
1889:
1853:
1740:
1578:
1435:
1366:
1339:
1308:
1281:
1242:
1215:
1136:
1033:
1006:
922:
894:
862:
678:
515:
484:
449:
338:
307:
247:
181:
161:
141:
2263:Transactions of the American Mathematical Society
1436:{\displaystyle p_{k}=x_{1}^{k}+\cdots +x_{n}^{k}}
149:be a ring of formal power series in the variable
2300:"The Poincare Polynomial of a Symmetric Product"
83:or power series, such as the number of integer
1751:
110:Definition, main properties and basic examples
8:
1945:{\displaystyle \operatorname {Sym} ^{n}(X)}
87:. It is also an important technique in the
2193:. In the case of quiver gauge theories of
1752:Macdonald's formula for symmetric products
1374:are related to the power sum polynomials:
169:, with coefficients in a commutative ring
2430:
2373:
2274:
2162:
2157:
2125:
2120:
2104:
2080:
2064:
2050:
2036:
2014:
2003:
1989:
1971:
1959:
1957:
1924:
1918:
1872:
1866:
1845:
1840:
1825:
1815:
1804:
1782:
1776:
1726:
1704:
1689:
1679:
1673:
1658:
1649:
1644:
1638:
1628:
1609:
1598:
1592:
1564:
1542:
1530:
1520:
1514:
1502:
1493:
1488:
1482:
1472:
1461:
1455:
1427:
1422:
1403:
1398:
1385:
1379:
1358:
1352:
1331:
1325:
1300:
1294:
1273:
1267:
1252:complete homogeneous symmetric polynomial
1234:
1228:
1216:{\displaystyle x_{1},x_{2},\ldots ,x_{n}}
1207:
1188:
1175:
1169:
1126:
1118:
1108:
1089:
1078:
1048:
1046:
1025:
1019:
998:
988:
978:
967:
946:
916:
915:
907:
878:
850:
822:
783:
774:
766:
765:
747:
731:
715:
703:
699:
697:
663:
648:
624:
606:
592:
565:
537:
533:
531:
499:
497:
465:
427:
414:
408:
397:
356:
354:
322:
320:
284:
263:
203:
197:
174:
154:
119:
1952:, is obtained from the series expansion:
1041:, then it is not difficult to show that:
95:, and many other combinatorial objects.
2214:
1905:. Then the Poincaré polynomial of the
60:, defined in the context of so-called
7:
2181:The plethystic programme in physics
40:, as a concise relation between the
2228:. New York, NY: Springer New York.
75:for many well studied sequences of
1610:
1473:
1090:
979:
523:when the variable is understood):
409:
71:, the plethystic exponential is a
14:
2353:Florentino, Carlos (2021-10-07).
2276:10.1090/S0002-9947-1955-0068198-2
1160:Relation with symmetric functions
923:{\displaystyle n\in \mathbb {N} }
1318:elementary symmetric polynomials
248:{\displaystyle R^{0}]\subset R]}
2222:PĂłlya, G.; Read, R. C. (1987).
2419:Journal of High Energy Physics
2154:
2145:
2140:
2134:
2092:
2086:
2061:
2051:
1993:
1986:
1977:
1964:
1939:
1933:
1884:
1878:
1837:
1831:
1794:
1788:
1735:
1694:
1683:
1663:
1625:
1615:
1573:
1535:
1524:
1507:
1115:
1095:
1068:
1062:
1059:
1053:
957:
951:
889:
883:
843:
837:
721:
708:
660:
653:
638:
629:
617:
611:
603:
597:
582:
570:
548:
542:
510:
504:
479:
473:
433:
420:
376:
370:
367:
361:
333:
327:
308:{\displaystyle f(x)\in R^{0}]}
302:
299:
293:
290:
274:
268:
242:
239:
233:
230:
221:
218:
212:
209:
136:
133:
127:
124:
1:
2449:10.1088/1126-6708/2007/03/090
2191:supersymmetric gauge theories
315:, its plethystic exponential
2362:Discrete Mathematics Letters
2257:Harary, Frank (1955-02-01).
485:{\displaystyle \exp(\cdot )}
902:is number of partitions of
516:{\displaystyle {\text{PE}}}
339:{\displaystyle {\text{PE}}}
2501:
2324:10.1017/S0305004100040573
2298:Macdonald, I. G. (1962).
2234:10.1007/978-1-4612-4664-0
1254:, that is the sum of all
689:Some basic examples are:
89:enumerative combinatorics
1909:th symmetric product of
1890:{\displaystyle b_{k}(X)}
1164:Working with variables
2172:
2019:
1946:
1891:
1855:
1820:
1742:
1614:
1580:
1477:
1437:
1368:
1341:
1310:
1283:
1244:
1217:
1138:
1094:
1035:
1008:
983:
924:
896:
873:In this last example,
864:
680:
517:
486:
451:
413:
340:
309:
249:
183:
163:
143:
32:which, like the usual
22:plethystic exponential
2384:10.47443/dml.2021.094
2173:
1999:
1947:
1892:
1856:
1800:
1743:
1594:
1581:
1457:
1438:
1369:
1367:{\displaystyle e_{k}}
1342:
1340:{\displaystyle h_{k}}
1311:
1309:{\displaystyle e_{k}}
1284:
1282:{\displaystyle x_{i}}
1245:
1243:{\displaystyle h_{k}}
1218:
1139:
1074:
1036:
1034:{\displaystyle a_{k}}
1009:
963:
925:
897:
865:
681:
518:
487:
452:
393:
341:
310:
250:
184:
164:
144:
2205:of the singularity.
1956:
1917:
1865:
1775:
1591:
1454:
1378:
1351:
1324:
1293:
1266:
1227:
1168:
1045:
1018:
945:
906:
895:{\displaystyle p(n)}
877:
696:
530:
496:
464:
353:
319:
262:
196:
173:
153:
118:
34:exponential function
28:defined on (formal)
2485:Symmetric functions
2441:2007JHEP...03..090F
2316:1962PCPS...58..563M
1770:Poincaré polynomial
1445:Newton's identities
1432:
1408:
937:Product-sum formula
73:generating function
38:symmetric functions
2168:
2115:
1942:
1887:
1851:
1738:
1576:
1433:
1418:
1394:
1364:
1337:
1306:
1279:
1240:
1213:
1146:integer partitions
1134:
1031:
1004:
920:
892:
860:
858:
833:
676:
674:
513:
482:
447:
336:
305:
245:
179:
159:
142:{\displaystyle R]}
139:
2243:978-1-4612-9105-3
2100:
1962:
1692:
1661:
1533:
1505:
1262:in the variables
1051:
818:
799:
777:
754:
706:
651:
627:
609:
595:
568:
540:
502:
440:
359:
325:
182:{\displaystyle R}
162:{\displaystyle x}
42:generating series
2492:
2469:
2468:
2434:
2410:
2404:
2403:
2377:
2359:
2350:
2344:
2343:
2295:
2289:
2288:
2278:
2254:
2248:
2247:
2219:
2177:
2175:
2174:
2169:
2167:
2166:
2144:
2143:
2130:
2129:
2114:
2096:
2095:
2085:
2084:
2075:
2074:
2049:
2045:
2041:
2040:
2018:
2013:
1976:
1975:
1963:
1960:
1951:
1949:
1948:
1943:
1929:
1928:
1896:
1894:
1893:
1888:
1877:
1876:
1860:
1858:
1857:
1852:
1850:
1849:
1830:
1829:
1819:
1814:
1787:
1786:
1747:
1745:
1744:
1739:
1731:
1730:
1709:
1708:
1693:
1690:
1678:
1677:
1662:
1659:
1654:
1653:
1643:
1642:
1633:
1632:
1613:
1608:
1585:
1583:
1582:
1577:
1569:
1568:
1547:
1546:
1534:
1531:
1519:
1518:
1506:
1503:
1498:
1497:
1487:
1486:
1476:
1471:
1442:
1440:
1439:
1434:
1431:
1426:
1407:
1402:
1390:
1389:
1373:
1371:
1370:
1365:
1363:
1362:
1346:
1344:
1343:
1338:
1336:
1335:
1315:
1313:
1312:
1307:
1305:
1304:
1288:
1286:
1285:
1280:
1278:
1277:
1249:
1247:
1246:
1241:
1239:
1238:
1222:
1220:
1219:
1214:
1212:
1211:
1193:
1192:
1180:
1179:
1143:
1141:
1140:
1135:
1133:
1132:
1131:
1130:
1113:
1112:
1093:
1088:
1052:
1049:
1040:
1038:
1037:
1032:
1030:
1029:
1013:
1011:
1010:
1005:
1003:
1002:
993:
992:
982:
977:
929:
927:
926:
921:
919:
901:
899:
898:
893:
869:
867:
866:
861:
859:
855:
854:
832:
804:
800:
798:
784:
778:
775:
769:
755:
753:
752:
751:
732:
720:
719:
707:
704:
685:
683:
682:
677:
675:
671:
670:
652:
649:
628:
625:
610:
607:
596:
593:
569:
566:
541:
538:
522:
520:
519:
514:
503:
500:
491:
489:
488:
483:
456:
454:
453:
448:
446:
442:
441:
436:
432:
431:
415:
412:
407:
360:
357:
345:
343:
342:
337:
326:
323:
314:
312:
311:
306:
289:
288:
254:
252:
251:
246:
208:
207:
188:
186:
185:
180:
168:
166:
165:
160:
148:
146:
145:
140:
2500:
2499:
2495:
2494:
2493:
2491:
2490:
2489:
2475:
2474:
2473:
2472:
2412:
2411:
2407:
2357:
2352:
2351:
2347:
2297:
2296:
2292:
2256:
2255:
2251:
2244:
2221:
2220:
2216:
2211:
2183:
2158:
2121:
2116:
2076:
2060:
2032:
2025:
2021:
2020:
1967:
1954:
1953:
1920:
1915:
1914:
1868:
1863:
1862:
1841:
1821:
1778:
1773:
1772:
1764:, of dimension
1754:
1722:
1700:
1669:
1645:
1634:
1624:
1589:
1588:
1560:
1538:
1510:
1489:
1478:
1452:
1451:
1381:
1376:
1375:
1354:
1349:
1348:
1327:
1322:
1321:
1296:
1291:
1290:
1269:
1264:
1263:
1258:of degree
1230:
1225:
1224:
1203:
1184:
1171:
1166:
1165:
1162:
1154:symmetric group
1122:
1114:
1104:
1043:
1042:
1021:
1016:
1015:
994:
984:
943:
942:
939:
904:
903:
875:
874:
857:
856:
846:
805:
788:
779:
771:
770:
743:
736:
724:
711:
694:
693:
673:
672:
659:
641:
621:
620:
585:
562:
561:
551:
528:
527:
494:
493:
462:
461:
423:
416:
392:
388:
351:
350:
317:
316:
280:
260:
259:
199:
194:
193:
171:
170:
151:
150:
116:
115:
112:
12:
11:
5:
2498:
2496:
2488:
2487:
2477:
2476:
2471:
2470:
2432:hep-th/0701063
2405:
2345:
2310:(4): 563–568.
2290:
2269:(2): 445–463.
2249:
2242:
2213:
2212:
2210:
2207:
2203:Hilbert series
2182:
2179:
2165:
2161:
2156:
2153:
2150:
2147:
2142:
2139:
2136:
2133:
2128:
2124:
2119:
2113:
2110:
2107:
2103:
2099:
2094:
2091:
2088:
2083:
2079:
2073:
2070:
2067:
2063:
2059:
2056:
2053:
2048:
2044:
2039:
2035:
2031:
2028:
2024:
2017:
2012:
2009:
2006:
2002:
1998:
1995:
1992:
1988:
1985:
1982:
1979:
1974:
1970:
1966:
1941:
1938:
1935:
1932:
1927:
1923:
1886:
1883:
1880:
1875:
1871:
1848:
1844:
1839:
1836:
1833:
1828:
1824:
1818:
1813:
1810:
1807:
1803:
1799:
1796:
1793:
1790:
1785:
1781:
1753:
1750:
1749:
1748:
1737:
1734:
1729:
1725:
1721:
1718:
1715:
1712:
1707:
1703:
1699:
1696:
1688:
1685:
1682:
1676:
1672:
1668:
1665:
1657:
1652:
1648:
1641:
1637:
1631:
1627:
1623:
1620:
1617:
1612:
1607:
1604:
1601:
1597:
1586:
1575:
1572:
1567:
1563:
1559:
1556:
1553:
1550:
1545:
1541:
1537:
1529:
1526:
1523:
1517:
1513:
1509:
1501:
1496:
1492:
1485:
1481:
1475:
1470:
1467:
1464:
1460:
1430:
1425:
1421:
1417:
1414:
1411:
1406:
1401:
1397:
1393:
1388:
1384:
1361:
1357:
1334:
1330:
1303:
1299:
1276:
1272:
1237:
1233:
1210:
1206:
1202:
1199:
1196:
1191:
1187:
1183:
1178:
1174:
1161:
1158:
1129:
1125:
1121:
1117:
1111:
1107:
1103:
1100:
1097:
1092:
1087:
1084:
1081:
1077:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1028:
1024:
1001:
997:
991:
987:
981:
976:
973:
970:
966:
962:
959:
956:
953:
950:
938:
935:
918:
914:
911:
891:
888:
885:
882:
871:
870:
853:
849:
845:
842:
839:
836:
831:
828:
825:
821:
817:
814:
811:
808:
806:
803:
797:
794:
791:
787:
782:
773:
772:
768:
764:
761:
758:
750:
746:
742:
739:
735:
730:
727:
725:
723:
718:
714:
710:
702:
701:
687:
686:
669:
666:
662:
658:
655:
647:
644:
642:
640:
637:
634:
631:
623:
622:
619:
616:
613:
605:
602:
599:
591:
588:
586:
584:
581:
578:
575:
572:
564:
563:
560:
557:
554:
552:
550:
547:
544:
536:
535:
512:
509:
506:
481:
478:
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472:
469:
458:
457:
445:
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279:
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256:
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214:
211:
206:
202:
178:
158:
138:
135:
132:
129:
126:
123:
111:
108:
91:of unlabelled
13:
10:
9:
6:
4:
3:
2:
2497:
2486:
2483:
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2253:
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2227:
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2206:
2204:
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2178:
2163:
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2042:
2037:
2033:
2029:
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2015:
2010:
2007:
2004:
2000:
1996:
1990:
1983:
1980:
1972:
1968:
1936:
1930:
1925:
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1912:
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1869:
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1105:
1101:
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733:
728:
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690:
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664:
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645:
643:
635:
632:
614:
600:
589:
587:
579:
576:
573:
558:
555:
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470:
467:
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424:
417:
404:
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398:
394:
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385:
382:
379:
373:
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348:
347:
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296:
285:
281:
277:
271:
265:
236:
227:
224:
215:
204:
200:
192:
191:
190:
189:. Denote by
176:
156:
130:
121:
109:
107:
105:
101:
96:
94:
90:
86:
82:
78:
74:
70:
69:combinatorics
65:
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
24:is a certain
23:
19:
2422:
2418:
2408:
2365:
2361:
2348:
2307:
2303:
2293:
2266:
2262:
2252:
2224:
2217:
2184:
1910:
1906:
1903:Betti number
1898:
1765:
1760:be a finite
1757:
1755:
1320:. Then, the
1259:
1223:, denote by
1163:
940:
932:
872:
688:
459:
346:is given by
257:
113:
97:
66:
62:lambda rings
30:power series
21:
15:
2187:Yang-Hui He
1150:cycle index
1148:and to the
81:polynomials
18:mathematics
2425:(3): 090.
2375:2105.13049
2209:References
2199:Calabi–Yau
1913:, denoted
1762:CW complex
1289:, and by
85:partitions
54:power sums
46:elementary
2457:1029-8479
2400:237451072
2392:2664-2557
2368:: 22–29.
2340:121316624
2332:0305-0041
2285:0002-9947
2149:−
2132:
2109:≥
2102:∑
2055:−
2030:−
2001:∏
1981:−
1931:
1802:∑
1720:−
1717:⋯
1714:−
1698:−
1667:−
1619:−
1611:∞
1596:∑
1555:⋯
1474:∞
1459:∑
1413:⋯
1256:monomials
1198:…
1120:−
1102:−
1091:∞
1076:∏
980:∞
965:∑
913:∈
827:≥
820:∑
793:−
763:∈
741:−
665:−
633:−
477:⋅
471:
410:∞
395:∑
386:
278:∈
225:⊂
2479:Category
2197:probing
2195:D-branes
1347:and the
104:topology
100:geometry
77:integers
58:plethysm
50:complete
26:operator
2465:1908174
2437:Bibcode
2312:Bibcode
1897:is its
1768:, with
1152:of the
2463:
2455:
2398:
2390:
2338:
2330:
2283:
2240:
1861:where
460:where
93:graphs
20:, the
2461:S2CID
2427:arXiv
2396:S2CID
2370:arXiv
2358:(PDF)
2336:S2CID
2453:ISSN
2423:2007
2388:ISSN
2328:ISSN
2281:ISSN
2238:ISBN
1756:Let
1316:the
1250:the
114:Let
102:and
52:and
44:for
2445:doi
2380:doi
2320:doi
2271:doi
2230:doi
2123:Sym
1922:Sym
1901:th
1443:by
468:exp
383:exp
98:In
67:In
64:.
16:In
2481::
2459:.
2451:.
2443:.
2435:.
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2334:.
2326:.
2318:.
2308:58
2306:.
2302:.
2279:.
2267:78
2265:.
2261:.
2236:.
1961:PE
1691:PE
1660:PE
1532:PE
1504:PE
1156:.
1050:PE
930:.
776:PE
705:PE
650:PE
626:PE
608:PE
594:PE
567:PE
539:PE
501:PE
358:PE
324:PE
79:,
48:,
2467:.
2447::
2439::
2429::
2402:.
2382::
2372::
2366:8
2342:.
2322::
2314::
2287:.
2273::
2246:.
2232::
2164:n
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2152:t
2146:(
2141:)
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2135:(
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2016:d
2011:0
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1997:=
1994:]
1991:x
1987:)
1984:t
1978:(
1973:X
1969:P
1965:[
1940:)
1937:X
1934:(
1926:n
1911:X
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654:[
646:=
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636:f
630:[
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615:g
612:[
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598:[
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580:g
577:+
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546:0
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421:(
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390:(
380:=
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374:x
371:(
368:]
365:f
362:[
334:]
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328:[
303:]
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297:x
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269:(
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231:[
228:R
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216:x
213:[
210:[
205:0
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177:R
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137:]
134:]
131:x
128:[
125:[
122:R
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