1873:
1434:
951:
1868:{\displaystyle {\begin{aligned}p_{n}&=P(x)={\frac {1-{\sqrt {1-4x}}}{2}}\\&={\frac {1}{2}}-{\frac {1}{2}}{\sqrt {1-4x}}\\&=-{\frac {1}{2}}\sum _{k=0}^{\infty }{{\frac {1}{2}} \choose k}(-4x)^{k}\\&=-{\frac {1}{2}}{{\frac {1}{2}} \choose n}(-4)^{n}\\&={\frac {1}{n}}{2n-2 \choose n-1}\end{aligned}}}
1274:. Putting the above description in words: A plane tree consists of a node to which is attached an arbitrary number of subtrees, each of which is also a plane tree. Using the operation on families of combinatorial structures developed earlier, this translates to a recursive generating function:
1109:
and cycles. A combinatorial structure is composed of atoms. For example, with trees the atoms would be the nodes. The atoms which compose the object can either be labeled or unlabeled. Unlabeled atoms are indistinguishable from each other, while labelled atoms are distinct. Therefore, for a
820:
1122:. There is generally a node called the root, which has no parent node. In plane trees each node can have an arbitrary number of children. In binary trees, a special case of plane trees, each node can have either two or no children. Let
1207:
1439:
946:{\displaystyle {\mbox{Seq}}({\mathcal {F}})=\epsilon \ \cup \ {\mathcal {F}}\ \cup \ {\mathcal {F}}\!\times \!{\mathcal {F}}\ \cup \ {\mathcal {F}}\!\times \!{\mathcal {F}}\!\times \!{\mathcal {F}}\ \cup \cdots }
605:
1087:
610:
Once determined, the generating function yields the information given by the previous approaches. In addition, the various natural operations on generating functions such as addition, multiplication,
444:
784:
1415:
722:
1338:
252:
278:
956:
To put the above in words: An empty sequence or a sequence of one element or a sequence of two elements or a sequence of three elements, etc. The generating function would be:
326:
1272:
1144:
668:
644:
360:
1236:
140:
Often, a complicated closed formula yields little insight into the behavior of the counting function as the number of counted objects grows. In these cases, a simple
513:
474:
200:
171:
1152:
1118:
Binary and plane trees are examples of an unlabeled combinatorial structure. Trees consist of nodes linked by edges in such a way that there are no
1907:
814:
generalizes the idea of the pair as defined above. Sequences are arbitrary
Cartesian products of a combinatorial object with itself. Formally:
525:
2087:
614:, etc., have a combinatorial significance; this allows one to extend results from one combinatorial problem in order to solve others.
962:
515:. Some common operation on families of combinatorial objects and its effect on the generating function will now be developed. The
1903:
376:
26:
that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting
755:
1357:
2233:
2107:
693:
2206:
2140:
2132:
2077:
2057:
1985:
1110:
combinatorial object consisting of labeled atoms a new object can be formed by simply swapping two or more atoms.
1990:
1280:
516:
2168:
2099:
205:
2174:
1995:
257:
2124:
2017:
283:
1970:
1253:
1125:
649:
625:
341:
109:, and so on. For instance, as shown below, the number of different possible orderings of a deck of
1975:
1955:
1215:
1965:
1950:
1146:
denote the family of all plane trees. Then this family can be recursively defined as follows:
1980:
126:
2045:
1119:
491:
452:
69:
65:
2216:
176:
147:
8:
2052:, Volumes 1 and 2. Elsevier (North-Holland), Amsterdam, and MIT Press, Cambridge, Mass.
1960:
1098:
1097:
The above operations can now be used to enumerate common combinatorial objects including
335:
141:
134:
130:
2041:
2160:
2113:
2026:
89:
16:
Area of combinatorics that deals with the number of ways certain patterns can be formed
1238:
represents the family of objects consisting of one node. This has generating function
2202:
2164:
2136:
2128:
2103:
2095:
2073:
2053:
749:
2022:
2212:
2150:
2013:
2196:
2030:
1202:{\displaystyle {\mathcal {P}}=\{\bullet \}\times \,{\mbox{Seq}}({\mathcal {P}})}
1918:
687:
106:
97:
77:
44:
2227:
2069:
2037:
73:
23:
2192:
2000:
72:, many of the problems that arise in applications have a relatively simple
101:, which can be expressed as a composition of elementary functions such as
2154:
2146:
1899:
1102:
485:
85:
81:
31:
27:
611:
2117:
1106:
102:
811:
600:{\displaystyle F(x)=\sum _{n=0}^{\infty }f_{n}{\frac {x^{n}}{n!}}}
1906:. To get from the fourth to fifth line manipulations using the
34:. More generally, given an infinite collection of finite sets
1082:{\displaystyle 1+F(x)+^{2}+^{3}+\cdots ={\frac {1}{1-F(x)}}.}
519:
is also sometimes used. In this case it would have the form
338:
are used to describe families of combinatorial objects. Let
2066:
The Crest of the
Peacock: Non-European Roots of Mathematics
2091:
1420:
An explicit formula for the number of plane trees of size
125:!. The problem of finding a closed formula is known as
1424:
can now be determined by extracting the coefficient of
1180:
825:
1437:
1360:
1283:
1256:
1218:
1155:
1128:
965:
823:
758:
696:
652:
628:
528:
494:
455:
379:
344:
286:
260:
208:
179:
150:
137:
and using this to arrive at the desired closed form.
476:
denotes the number of combinatorial objects of size
439:{\displaystyle F(x)=\sum _{n=0}^{\infty }f_{n}x^{n}}
779:{\displaystyle {\mathcal {F}}\times {\mathcal {G}}}
1867:
1410:{\displaystyle P(x)={\frac {1-{\sqrt {1-4x}}}{2}}}
1409:
1332:
1266:
1230:
1201:
1138:
1081:
945:
778:
716:
662:
638:
599:
507:
468:
438:
354:
320:
272:
246:
194:
165:
68:the number of elements in a set is a rather broad
1913:The expression on the last line is equal to the (
1855:
1823:
1775:
1755:
1701:
1681:
926:
922:
914:
910:
886:
882:
717:{\displaystyle {\mathcal {F}}\cup {\mathcal {G}}}
2225:
47:, enumerative combinatorics seeks to describe a
1904:Newton's generalization of the binomial theorem
480:. The number of combinatorial objects of size
144:approximation may be preferable. A function
2201:(2nd ed.). Boston, MA: Academic Press.
1225:
1219:
1172:
1166:
748:For two combinatorial families as above the
2188:, Wiley & Sons, New York (republished).
2181:, Wiley & Sons, New York (republished).
1333:{\displaystyle P(x)=x\,{\frac {1}{1-P(x)}}}
1092:
80:provides a unified framework for counting
2179:An Introduction to Combinatorial Analysis
1302:
1178:
1113:
317:
2018:Enumerative and Algebraic Combinatorics
330:
2226:
2063:
247:{\displaystyle f(n)/g(n)\rightarrow 1}
51:which counts the number of objects in
362:denote the family of objects and let
129:, and frequently involves deriving a
2191:
2064:Joseph, George Gheverghese (1994) .
273:{\displaystyle n\rightarrow \infty }
370:) be its generating function. Then
13:
1827:
1759:
1685:
1673:
1259:
1191:
1158:
1131:
929:
917:
905:
889:
877:
861:
836:
771:
761:
709:
699:
655:
631:
622:Given two combinatorial families,
560:
411:
347:
267:
173:is an asymptotic approximation to
14:
2245:
1250:) denote the generating function
1908:generalized binomial coefficient
321:{\displaystyle f(n)\sim g(n).\,}
95:The simplest such functions are
1991:Method of distinguished element
1898:). The series expansion of the
1886:) refers to the coefficient of
517:exponential generating function
1791:
1781:
1720:
1707:
1654:
1641:
1589:
1576:
1560:
1547:
1503:
1490:
1484:
1478:
1472:
1459:
1370:
1364:
1324:
1318:
1293:
1287:
1267:{\displaystyle {\mathcal {P}}}
1196:
1186:
1139:{\displaystyle {\mathcal {P}}}
1070:
1064:
1031:
1027:
1021:
1015:
1003:
999:
993:
987:
981:
975:
841:
831:
663:{\displaystyle {\mathcal {G}}}
639:{\displaystyle {\mathcal {F}}}
538:
532:
389:
383:
355:{\displaystyle {\mathcal {F}}}
311:
305:
296:
290:
264:
238:
235:
229:
218:
212:
189:
183:
160:
154:
1:
2007:
1986:Inclusion–exclusion principle
2086:Loehr, Nicholas A. (2011).
1231:{\displaystyle \{\bullet \}}
805:
752:(pair) of the two families (
7:
1943:
10:
2250:
2125:Cambridge University Press
2032:A Combinatorial Miscellany
786:) has generating function
724:) has generating function
670:with generating functions
484:is therefore given by the
2234:Enumerative combinatorics
2156:Combinatorial Enumeration
2119:Enumerative Combinatorics
2050:Handbook of Combinatorics
1996:Pólya enumeration theorem
1971:Combinatorial game theory
280:. In this case, we write
20:Enumerative combinatorics
2186:Combinatorial Identities
2068:(2nd ed.). London:
1976:Combinatorial principles
1956:Asymptotic combinatorics
1093:Combinatorial structures
743:
617:
2198:Generatingfunctionology
2088:Bijective Combinatorics
1966:Combinatorial explosion
1951:Algebraic combinatorics
2184:Riordan, John (1968).
1917: − 1)
1869:
1677:
1411:
1334:
1268:
1232:
1203:
1140:
1114:Binary and plane trees
1083:
947:
780:
718:
664:
640:
601:
564:
509:
470:
440:
415:
356:
322:
274:
248:
196:
167:
1981:Combinatorial species
1870:
1657:
1412:
1335:
1269:
1233:
1204:
1141:
1084:
948:
781:
719:
690:of the two families (
665:
641:
602:
544:
510:
508:{\displaystyle x^{n}}
471:
469:{\displaystyle f_{n}}
441:
395:
357:
323:
275:
249:
197:
168:
127:algebraic enumeration
1878:Note: The notation
1435:
1358:
1281:
1254:
1216:
1153:
1126:
963:
821:
756:
694:
686:) respectively, the
650:
626:
526:
492:
453:
377:
342:
336:Generating functions
331:Generating functions
284:
258:
206:
195:{\displaystyle f(n)}
177:
166:{\displaystyle g(n)}
148:
70:mathematical problem
2114:Stanley, Richard P.
2027:Stanley, Richard P.
135:generating function
131:recurrence relation
2161:Dover Publications
1865:
1863:
1407:
1343:After solving for
1330:
1264:
1228:
1199:
1184:
1136:
1079:
943:
829:
776:
714:
660:
636:
597:
505:
466:
436:
352:
318:
270:
244:
192:
163:
2151:Jackson, David M.
2121:, Volumes 1 and 2
2038:Graham, Ronald L.
2014:Zeilberger, Doron
1853:
1818:
1773:
1768:
1750:
1699:
1694:
1639:
1616:
1600:
1571:
1535:
1529:
1405:
1399:
1328:
1183:
1074:
936:
902:
896:
874:
868:
858:
852:
828:
750:Cartesian product
595:
76:description. The
49:counting function
2241:
2220:
2193:Wilf, Herbert S.
2083:
1961:Burnside's lemma
1874:
1872:
1871:
1866:
1864:
1860:
1859:
1858:
1852:
1841:
1826:
1819:
1811:
1803:
1799:
1798:
1780:
1779:
1778:
1769:
1761:
1758:
1751:
1743:
1732:
1728:
1727:
1706:
1705:
1704:
1695:
1687:
1684:
1676:
1671:
1653:
1652:
1640:
1632:
1621:
1617:
1603:
1601:
1593:
1588:
1587:
1572:
1564:
1559:
1558:
1540:
1536:
1531:
1530:
1516:
1507:
1502:
1501:
1471:
1470:
1451:
1450:
1416:
1414:
1413:
1408:
1406:
1401:
1400:
1386:
1377:
1339:
1337:
1336:
1331:
1329:
1327:
1304:
1273:
1271:
1270:
1265:
1263:
1262:
1237:
1235:
1234:
1229:
1208:
1206:
1205:
1200:
1195:
1194:
1185:
1181:
1162:
1161:
1145:
1143:
1142:
1137:
1135:
1134:
1088:
1086:
1085:
1080:
1075:
1073:
1050:
1039:
1038:
1011:
1010:
952:
950:
949:
944:
934:
933:
932:
921:
920:
909:
908:
900:
894:
893:
892:
881:
880:
872:
866:
865:
864:
856:
850:
840:
839:
830:
826:
785:
783:
782:
777:
775:
774:
765:
764:
723:
721:
720:
715:
713:
712:
703:
702:
669:
667:
666:
661:
659:
658:
645:
643:
642:
637:
635:
634:
606:
604:
603:
598:
596:
594:
586:
585:
576:
574:
573:
563:
558:
514:
512:
511:
506:
504:
503:
475:
473:
472:
467:
465:
464:
445:
443:
442:
437:
435:
434:
425:
424:
414:
409:
361:
359:
358:
353:
351:
350:
327:
325:
324:
319:
279:
277:
276:
271:
253:
251:
250:
245:
225:
201:
199:
198:
193:
172:
170:
169:
164:
2249:
2248:
2244:
2243:
2242:
2240:
2239:
2238:
2224:
2223:
2209:
2147:Goulden, Ian P.
2116:(1997, 1999).
2080:
2048:, eds. (1996).
2023:Björner, Anders
2010:
2005:
1946:
1939:
1929:
1862:
1861:
1854:
1842:
1828:
1822:
1821:
1820:
1810:
1801:
1800:
1794:
1790:
1774:
1760:
1754:
1753:
1752:
1742:
1730:
1729:
1723:
1719:
1700:
1686:
1680:
1679:
1678:
1672:
1661:
1648:
1644:
1631:
1619:
1618:
1602:
1592:
1583:
1579:
1563:
1554:
1550:
1538:
1537:
1515:
1508:
1506:
1497:
1493:
1466:
1462:
1452:
1446:
1442:
1438:
1436:
1433:
1432:
1385:
1378:
1376:
1359:
1356:
1355:
1308:
1303:
1282:
1279:
1278:
1258:
1257:
1255:
1252:
1251:
1217:
1214:
1213:
1190:
1189:
1179:
1157:
1156:
1154:
1151:
1150:
1130:
1129:
1127:
1124:
1123:
1116:
1095:
1054:
1049:
1034:
1030:
1006:
1002:
964:
961:
960:
928:
927:
916:
915:
904:
903:
888:
887:
876:
875:
860:
859:
835:
834:
824:
822:
819:
818:
808:
770:
769:
760:
759:
757:
754:
753:
746:
708:
707:
698:
697:
695:
692:
691:
654:
653:
651:
648:
647:
630:
629:
627:
624:
623:
620:
612:differentiation
587:
581:
577:
575:
569:
565:
559:
548:
527:
524:
523:
499:
495:
493:
490:
489:
460:
456:
454:
451:
450:
430:
426:
420:
416:
410:
399:
378:
375:
374:
346:
345:
343:
340:
339:
333:
285:
282:
281:
259:
256:
255:
221:
207:
204:
203:
178:
175:
174:
149:
146:
145:
98:closed formulas
59:
45:natural numbers
43:indexed by the
42:
17:
12:
11:
5:
2247:
2237:
2236:
2222:
2221:
2207:
2189:
2182:
2172:
2144:
2111:
2108:978-1439848845
2084:
2078:
2061:
2046:Lovász, László
2035:
2020:
2009:
2006:
2004:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1947:
1945:
1942:
1934:
1925:
1919:Catalan number
1876:
1875:
1857:
1851:
1848:
1845:
1840:
1837:
1834:
1831:
1825:
1817:
1814:
1809:
1806:
1804:
1802:
1797:
1793:
1789:
1786:
1783:
1777:
1772:
1767:
1764:
1757:
1749:
1746:
1741:
1738:
1735:
1733:
1731:
1726:
1722:
1718:
1715:
1712:
1709:
1703:
1698:
1693:
1690:
1683:
1675:
1670:
1667:
1664:
1660:
1656:
1651:
1647:
1643:
1638:
1635:
1630:
1627:
1624:
1622:
1620:
1615:
1612:
1609:
1606:
1599:
1596:
1591:
1586:
1582:
1578:
1575:
1570:
1567:
1562:
1557:
1553:
1549:
1546:
1543:
1541:
1539:
1534:
1528:
1525:
1522:
1519:
1514:
1511:
1505:
1500:
1496:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1469:
1465:
1461:
1458:
1455:
1453:
1449:
1445:
1441:
1440:
1418:
1417:
1404:
1398:
1395:
1392:
1389:
1384:
1381:
1375:
1372:
1369:
1366:
1363:
1341:
1340:
1326:
1323:
1320:
1317:
1314:
1311:
1307:
1301:
1298:
1295:
1292:
1289:
1286:
1261:
1227:
1224:
1221:
1210:
1209:
1198:
1193:
1188:
1177:
1174:
1171:
1168:
1165:
1160:
1133:
1115:
1112:
1094:
1091:
1090:
1089:
1078:
1072:
1069:
1066:
1063:
1060:
1057:
1053:
1048:
1045:
1042:
1037:
1033:
1029:
1026:
1023:
1020:
1017:
1014:
1009:
1005:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
954:
953:
942:
939:
931:
925:
919:
913:
907:
899:
891:
885:
879:
871:
863:
855:
849:
846:
843:
838:
833:
807:
804:
773:
768:
763:
745:
742:
711:
706:
701:
688:disjoint union
657:
633:
619:
616:
608:
607:
593:
590:
584:
580:
572:
568:
562:
557:
554:
551:
547:
543:
540:
537:
534:
531:
502:
498:
463:
459:
447:
446:
433:
429:
423:
419:
413:
408:
405:
402:
398:
394:
391:
388:
385:
382:
349:
332:
329:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
269:
266:
263:
243:
240:
237:
234:
231:
228:
224:
220:
217:
214:
211:
191:
188:
185:
182:
162:
159:
156:
153:
78:twelvefold way
55:
38:
22:is an area of
15:
9:
6:
4:
3:
2:
2246:
2235:
2232:
2231:
2229:
2218:
2214:
2210:
2208:0-12-751956-4
2204:
2200:
2199:
2194:
2190:
2187:
2183:
2180:
2176:
2175:Riordan, John
2173:
2170:
2166:
2162:
2158:
2157:
2152:
2148:
2145:
2142:
2141:0-521-56069-1
2138:
2134:
2133:0-521-55309-1
2130:
2126:
2122:
2120:
2115:
2112:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2079:0-14-012529-9
2075:
2071:
2070:Penguin Books
2067:
2062:
2059:
2058:0-262-07169-X
2055:
2051:
2047:
2043:
2039:
2036:
2034:
2033:
2028:
2024:
2021:
2019:
2015:
2012:
2011:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1948:
1941:
1937:
1933:
1928:
1924:
1921:. Therefore,
1920:
1916:
1911:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1849:
1846:
1843:
1838:
1835:
1832:
1829:
1815:
1812:
1807:
1805:
1795:
1787:
1784:
1770:
1765:
1762:
1747:
1744:
1739:
1736:
1734:
1724:
1716:
1713:
1710:
1696:
1691:
1688:
1668:
1665:
1662:
1658:
1649:
1645:
1636:
1633:
1628:
1625:
1623:
1613:
1610:
1607:
1604:
1597:
1594:
1584:
1580:
1573:
1568:
1565:
1555:
1551:
1544:
1542:
1532:
1526:
1523:
1520:
1517:
1512:
1509:
1498:
1494:
1487:
1481:
1475:
1467:
1463:
1456:
1454:
1447:
1443:
1431:
1430:
1429:
1427:
1423:
1402:
1396:
1393:
1390:
1387:
1382:
1379:
1373:
1367:
1361:
1354:
1353:
1352:
1350:
1346:
1321:
1315:
1312:
1309:
1305:
1299:
1296:
1290:
1284:
1277:
1276:
1275:
1249:
1245:
1241:
1222:
1212:In this case
1175:
1169:
1163:
1149:
1148:
1147:
1121:
1111:
1108:
1104:
1100:
1076:
1067:
1061:
1058:
1055:
1051:
1046:
1043:
1040:
1035:
1024:
1018:
1012:
1007:
996:
990:
984:
978:
972:
969:
966:
959:
958:
957:
940:
937:
923:
911:
897:
883:
869:
853:
847:
844:
817:
816:
815:
813:
803:
801:
797:
793:
789:
766:
751:
741:
739:
735:
731:
727:
704:
689:
685:
681:
677:
673:
615:
613:
591:
588:
582:
578:
570:
566:
555:
552:
549:
545:
541:
535:
529:
522:
521:
520:
518:
500:
496:
487:
483:
479:
461:
457:
431:
427:
421:
417:
406:
403:
400:
396:
392:
386:
380:
373:
372:
371:
369:
365:
337:
328:
314:
308:
302:
299:
293:
287:
261:
241:
232:
226:
222:
215:
209:
186:
180:
157:
151:
143:
138:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
99:
93:
91:
87:
83:
79:
75:
74:combinatorial
71:
67:
63:
58:
54:
50:
46:
41:
37:
33:
30:and counting
29:
25:
24:combinatorics
21:
2197:
2185:
2178:
2155:
2118:
2065:
2049:
2042:Grötschel M.
2031:
2001:Sieve theory
1935:
1931:
1926:
1922:
1914:
1912:
1902:is based on
1895:
1891:
1887:
1883:
1879:
1877:
1425:
1421:
1419:
1348:
1344:
1342:
1247:
1243:
1239:
1211:
1117:
1105:and plane),
1096:
955:
809:
799:
795:
791:
787:
747:
737:
733:
729:
725:
683:
679:
675:
671:
621:
609:
481:
477:
448:
367:
363:
334:
139:
122:
118:
114:
110:
96:
94:
86:combinations
82:permutations
61:
56:
52:
48:
39:
35:
32:permutations
28:combinations
19:
18:
1910:is needed.
1900:square root
810:A (finite)
486:coefficient
64:. Although
2217:0831.05001
2169:0486435970
2100:143984884X
2008:References
1107:Dyck paths
142:asymptotic
103:factorials
90:partitions
2153:(2004).
2092:CRC Press
1847:−
1836:−
1785:−
1740:−
1711:−
1674:∞
1659:∑
1629:−
1608:−
1574:−
1521:−
1513:−
1391:−
1383:−
1313:−
1223:∙
1176:×
1170:∙
1059:−
1044:⋯
941:⋯
938:∪
924:×
912:×
898:∪
884:×
870:∪
854:∪
848:ϵ
806:Sequences
767:×
705:∪
561:∞
546:∑
412:∞
397:∑
300:∼
268:∞
265:→
239:→
113:cards is
60:for each
2228:Category
2195:(1994).
2177:(1958).
1944:See also
812:sequence
66:counting
2215:
2205:
2167:
2139:
2131:
2106:
2098:
2076:
2056:
2044:, and
1242:. Let
1120:cycles
1103:binary
935:
901:
895:
873:
867:
857:
851:
678:) and
449:where
107:powers
1099:trees
744:Pairs
618:Union
2203:ISBN
2165:ISBN
2149:and
2137:ISBN
2129:ISBN
2104:ISBN
2096:ISBN
2074:ISBN
2054:ISBN
2025:and
732:) +
646:and
121:) =
88:and
2213:Zbl
2163:.
2127:.
2102:,
2094:.
2090:.
2040:,
1890:in
1351:):
1182:Seq
827:Seq
802:).
740:).
488:of
254:as
202:if
133:or
2230::
2211:.
2159:.
2135:,
2123:.
2072:.
2029:,
2016:,
1940:.
1938:−1
1930:=
1428::
105:,
92:.
84:,
2219:.
2171:.
2143:.
2110:.
2082:.
2060:.
1936:n
1932:c
1927:n
1923:p
1915:n
1896:x
1894:(
1892:f
1888:x
1884:x
1882:(
1880:f
1856:)
1850:1
1844:n
1839:2
1833:n
1830:2
1824:(
1816:n
1813:1
1808:=
1796:n
1792:)
1788:4
1782:(
1776:)
1771:n
1766:2
1763:1
1756:(
1748:2
1745:1
1737:=
1725:k
1721:)
1717:x
1714:4
1708:(
1702:)
1697:k
1692:2
1689:1
1682:(
1669:0
1666:=
1663:k
1655:]
1650:n
1646:x
1642:[
1637:2
1634:1
1626:=
1614:x
1611:4
1605:1
1598:2
1595:1
1590:]
1585:n
1581:x
1577:[
1569:2
1566:1
1561:]
1556:n
1552:x
1548:[
1545:=
1533:2
1527:x
1524:4
1518:1
1510:1
1504:]
1499:n
1495:x
1491:[
1488:=
1485:)
1482:x
1479:(
1476:P
1473:]
1468:n
1464:x
1460:[
1457:=
1448:n
1444:p
1426:x
1422:n
1403:2
1397:x
1394:4
1388:1
1380:1
1374:=
1371:)
1368:x
1365:(
1362:P
1349:x
1347:(
1345:P
1325:)
1322:x
1319:(
1316:P
1310:1
1306:1
1300:x
1297:=
1294:)
1291:x
1288:(
1285:P
1260:P
1248:x
1246:(
1244:P
1240:x
1226:}
1220:{
1197:)
1192:P
1187:(
1173:}
1167:{
1164:=
1159:P
1132:P
1101:(
1077:.
1071:)
1068:x
1065:(
1062:F
1056:1
1052:1
1047:=
1041:+
1036:3
1032:]
1028:)
1025:x
1022:(
1019:F
1016:[
1013:+
1008:2
1004:]
1000:)
997:x
994:(
991:F
988:[
985:+
982:)
979:x
976:(
973:F
970:+
967:1
930:F
918:F
906:F
890:F
878:F
862:F
845:=
842:)
837:F
832:(
800:x
798:(
796:G
794:)
792:x
790:(
788:F
772:G
762:F
738:x
736:(
734:G
730:x
728:(
726:F
710:G
700:F
684:x
682:(
680:G
676:x
674:(
672:F
656:G
632:F
592:!
589:n
583:n
579:x
571:n
567:f
556:0
553:=
550:n
542:=
539:)
536:x
533:(
530:F
501:n
497:x
482:n
478:n
462:n
458:f
432:n
428:x
422:n
418:f
407:0
404:=
401:n
393:=
390:)
387:x
384:(
381:F
368:x
366:(
364:F
348:F
315:.
312:)
309:n
306:(
303:g
297:)
294:n
291:(
288:f
262:n
242:1
236:)
233:n
230:(
227:g
223:/
219:)
216:n
213:(
210:f
190:)
187:n
184:(
181:f
161:)
158:n
155:(
152:g
123:n
119:n
117:(
115:f
111:n
62:n
57:n
53:S
40:i
36:S
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