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Ewens's sampling formula

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22: 2104: 2114: 355: 160: 543: 412: 350:{\displaystyle \operatorname {Pr} (a_{1},\dots ,a_{n};\theta )={n! \over \theta (\theta +1)\cdots (\theta +n-1)}\prod _{j=1}^{n}{\theta ^{a_{j}} \over j^{a_{j}}a_{j}!},} 762: 420: 51: 2138: 891: 642:
gametes are chosen without replacement, then the resulting probability distribution on the set of all partitions of the smaller integer
2117: 1374: 1282: 2069: 1935: 1147: 906: 755: 1830: 1594: 680: 1268: 712:, 31:1 (Feb 2016). This article introduces a series of seven articles about Ewens Sampling in a special issue of the journal. 670: 1589: 1533: 1431: 1193: 831: 705: 1875: 1609: 1462: 1137: 881: 1339: 2107: 1779: 1755: 1334: 748: 1976: 1853: 1814: 1786: 1760: 1678: 1604: 1027: 775: 734: 73: 615: = 1, then the distribution is precisely that of the integer partition induced by a uniformly distributed 44: 1964: 1930: 1796: 1791: 1636: 1444: 1142: 896: 2148: 1714: 1627: 1599: 1508: 1457: 1329: 1112: 1077: 1728: 1645: 1482: 1406: 1229: 1107: 1082: 946: 941: 936: 722:
S. Tavare and W. J. Ewens, "The Multivariate Ewens distribution." (1997, Chapter 41 from the reference below).
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The phrase "under certain conditions" used above is made precise by the following assumptions:
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This family of probability distributions enjoys the property that if after the sample of
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Proceedings of the Royal Society of London, Series B, Mathematical and Physical Sciences
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and the role of selection at the locus in question is negligible; and
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Warren Ewens, "The sampling theory of selectively neutral alleles",
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is small by comparison to the size of the whole population; and
144: 122: 99: 591:. Among probabilists and statisticians it is often called the 715:
J.F.C. Kingman, "Random partitions in population genetics",
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is taken from a population and classified according to the
423: 374: 163: 770: 623: â†’ âˆž, the probability that no two of the 559:The population is in statistical equilibrium under 725:N.L. Johnson, S. Kotz, and N. Balakrishnan (1997) 537: 406: 349: 657:The Ewens distribution arises naturally from the 2130: 414:is a sequence of nonnegative integers such that 43:but its sources remain unclear because it lacks 607: = 0, the probability is 1 that all 756: 646:is just what the formula above would give if 98:associated with counts of how many different 102:are observed a given number of times in the 763: 749: 598: 534: 154:alleles represented twice, and so on, is 74:Learn how and when to remove this message 114:Ewens's sampling formula, introduced by 2131: 681:Unified neutral theory of biodiversity 627:genes are the same approaches 1. 744: 706:The Ubiquitous Ewens Sampling Formula 701:, volume 3, pages 87–112, 1972. 671:Chinese restaurant table distribution 2113: 147:represented once in the sample, and 15: 2139:Theory of probability distributions 727:Discrete Multivariate Distributions 407:{\displaystyle a_{1},\ldots ,a_{n}} 13: 14: 2160: 2112: 2103: 2102: 719:, volume 361, number 1704, 1978. 20: 593:multivariate Ewens distribution 699:Theoretical Population Biology 570:Every mutant allele is novel. 264: 246: 240: 228: 208: 170: 1: 109: 7: 664: 10: 2165: 1936:Wrapped asymmetric Laplace 907:Extended negative binomial 659:Chinese restaurant process 650:were put in place of  571: 2098: 2032: 1990: 1891: 1727: 1705: 1696: 1595:Generalized extreme value 1580: 1415: 1375:Relativistic Breit–Wigner 1091: 988: 979: 872: 792: 783: 772:Probability distributions 611:genes are the same. When 586:partitions of the integer 360:for some positive number 691: 582:probability distribution 366:population mutation rate 92:Ewens's sampling formula 29:This article includes a 1590:Generalized chi-squared 1534:Normal-inverse Gaussian 599:Mathematical properties 58:more precise citations. 2149:Discrete distributions 1902:Univariate (circular) 1463:Generalized hyperbolic 892:Conway–Maxwell–Poisson 882:Beta negative binomial 574:Infinite-alleles model 539: 511: 408: 351: 290: 1947:Bivariate (spherical) 1445:Kaniadakis Îș-Gaussian 540: 491: 409: 352: 270: 2012:Dirac delta function 1959:Bivariate (toroidal) 1916:Univariate von Mises 1787:Multivariate Laplace 1679:Shifted log-logistic 1028:Continuous Bernoulli 421: 372: 161: 2144:Population genetics 2060:Natural exponential 1965:Bivariate von Mises 1931:Wrapped exponential 1797:Multivariate stable 1792:Multivariate normal 1113:Benktander 2nd kind 1108:Benktander 1st kind 897:Discrete phase-type 710:Statistical Science 88:population genetics 1715:Rectified Gaussian 1600:Generalized Pareto 1458:Generalized normal 1330:Matrix-exponential 704:H. Crane. (2016) " 617:random permutation 584:on the set of all 535: 404: 347: 31:list of references 2126: 2125: 1723: 1722: 1692: 1691: 1583:whose type varies 1529:Normal (Gaussian) 1483:Hyperbolic secant 1432:Exponential power 1335:Maxwell–Boltzmann 1083:Wigner semicircle 975: 974: 947:Parabolic fractal 937:Negative binomial 676:Coalescent theory 364:representing the 342: 268: 84: 83: 76: 2156: 2116: 2115: 2106: 2105: 2045:Compound Poisson 2020: 2008: 1977:von Mises–Fisher 1973: 1961: 1949: 1911:Circular uniform 1907: 1827: 1771: 1742: 1703: 1702: 1605:Marchenko–Pastur 1468:Geometric stable 1385:Truncated normal 1278:Inverse Gaussian 1184:Hyperexponential 1023:Beta rectangular 991:bounded interval 986: 985: 854:Discrete uniform 839:Poisson binomial 790: 789: 765: 758: 751: 742: 741: 552:The sample size 544: 542: 541: 536: 524: 523: 510: 505: 487: 486: 465: 464: 449: 448: 433: 432: 413: 411: 410: 405: 403: 402: 384: 383: 356: 354: 353: 348: 343: 341: 337: 336: 327: 326: 325: 324: 309: 308: 307: 306: 292: 289: 284: 269: 267: 223: 215: 201: 200: 182: 181: 129:at a particular 79: 72: 68: 65: 59: 54:this article by 45:inline citations 24: 23: 16: 2164: 2163: 2159: 2158: 2157: 2155: 2154: 2153: 2129: 2128: 2127: 2122: 2094: 2070:Maximum entropy 2028: 2016: 2004: 1994: 1986: 1969: 1957: 1945: 1900: 1887: 1824:Matrix-valued: 1821: 1767: 1738: 1730: 1719: 1707: 1698: 1688: 1582: 1576: 1493: 1419: 1417: 1411: 1340:Maxwell–JĂŒttner 1189:Hypoexponential 1095: 1093: 1092:supported on a 1087: 1048:Noncentral beta 1008:Balding–Nichols 990: 989:supported on a 981: 971: 874: 868: 864:Zipf–Mandelbrot 794: 785: 779: 769: 694: 667: 601: 576: 519: 515: 506: 495: 482: 478: 460: 456: 444: 440: 428: 424: 422: 419: 418: 398: 394: 379: 375: 373: 370: 369: 332: 328: 320: 316: 315: 311: 310: 302: 298: 297: 293: 291: 285: 274: 224: 216: 214: 196: 192: 177: 173: 162: 159: 158: 153: 143: 137:that there are 112: 80: 69: 63: 60: 49: 35:related reading 25: 21: 12: 11: 5: 2162: 2152: 2151: 2146: 2141: 2124: 2123: 2121: 2120: 2110: 2099: 2096: 2095: 2093: 2092: 2087: 2082: 2077: 2072: 2067: 2065:Location–scale 2062: 2057: 2052: 2047: 2042: 2036: 2034: 2030: 2029: 2027: 2026: 2021: 2014: 2009: 2001: 1999: 1988: 1987: 1985: 1984: 1979: 1974: 1967: 1962: 1955: 1950: 1943: 1938: 1933: 1928: 1926:Wrapped Cauchy 1923: 1921:Wrapped normal 1918: 1913: 1908: 1897: 1895: 1889: 1888: 1886: 1885: 1884: 1883: 1878: 1876:Normal-inverse 1873: 1868: 1858: 1857: 1856: 1846: 1838: 1833: 1828: 1819: 1818: 1817: 1807: 1799: 1794: 1789: 1784: 1783: 1782: 1772: 1765: 1764: 1763: 1758: 1748: 1743: 1735: 1733: 1725: 1724: 1721: 1720: 1718: 1717: 1711: 1709: 1700: 1694: 1693: 1690: 1689: 1687: 1686: 1681: 1676: 1668: 1660: 1652: 1643: 1634: 1625: 1616: 1607: 1602: 1597: 1592: 1586: 1584: 1578: 1577: 1575: 1574: 1569: 1567:Variance-gamma 1564: 1559: 1551: 1546: 1541: 1536: 1531: 1526: 1518: 1513: 1512: 1511: 1501: 1496: 1491: 1485: 1480: 1475: 1470: 1465: 1460: 1455: 1447: 1442: 1434: 1429: 1423: 1421: 1413: 1412: 1410: 1409: 1407:Wilks's lambda 1404: 1403: 1402: 1392: 1387: 1382: 1377: 1372: 1367: 1362: 1357: 1352: 1347: 1345:Mittag-Leffler 1342: 1337: 1332: 1327: 1322: 1317: 1312: 1307: 1302: 1297: 1292: 1287: 1286: 1285: 1275: 1266: 1261: 1256: 1255: 1254: 1244: 1242:gamma/Gompertz 1239: 1238: 1237: 1232: 1222: 1217: 1212: 1211: 1210: 1198: 1197: 1196: 1191: 1186: 1176: 1175: 1174: 1164: 1159: 1154: 1153: 1152: 1151: 1150: 1140: 1130: 1125: 1120: 1115: 1110: 1105: 1099: 1097: 1094:semi-infinite 1089: 1088: 1086: 1085: 1080: 1075: 1070: 1065: 1060: 1055: 1050: 1045: 1040: 1035: 1030: 1025: 1020: 1015: 1010: 1005: 1000: 994: 992: 983: 977: 976: 973: 972: 970: 969: 964: 959: 954: 949: 944: 939: 934: 929: 924: 919: 914: 909: 904: 899: 894: 889: 884: 878: 876: 873:with infinite 870: 869: 867: 866: 861: 856: 851: 846: 841: 836: 835: 834: 827:Hypergeometric 824: 819: 814: 809: 804: 798: 796: 787: 781: 780: 768: 767: 760: 753: 745: 739: 738: 723: 720: 713: 702: 693: 690: 689: 688: 686:Biomathematics 683: 678: 673: 666: 663: 600: 597: 578: 577: 568: 557: 546: 545: 533: 530: 527: 522: 518: 514: 509: 504: 501: 498: 494: 490: 485: 481: 477: 474: 471: 468: 463: 459: 455: 452: 447: 443: 439: 436: 431: 427: 401: 397: 393: 390: 387: 382: 378: 358: 357: 346: 340: 335: 331: 323: 319: 314: 305: 301: 296: 288: 283: 280: 277: 273: 266: 263: 260: 257: 254: 251: 248: 245: 242: 239: 236: 233: 230: 227: 222: 219: 213: 210: 207: 204: 199: 195: 191: 188: 185: 180: 176: 172: 169: 166: 151: 141: 111: 108: 94:describes the 82: 81: 39:external links 28: 26: 19: 9: 6: 4: 3: 2: 2161: 2150: 2147: 2145: 2142: 2140: 2137: 2136: 2134: 2119: 2111: 2109: 2101: 2100: 2097: 2091: 2088: 2086: 2083: 2081: 2078: 2076: 2073: 2071: 2068: 2066: 2063: 2061: 2058: 2056: 2053: 2051: 2048: 2046: 2043: 2041: 2038: 2037: 2035: 2031: 2025: 2022: 2019: 2015: 2013: 2010: 2007: 2003: 2002: 2000: 1998: 1993: 1989: 1983: 1980: 1978: 1975: 1972: 1968: 1966: 1963: 1960: 1956: 1954: 1951: 1948: 1944: 1942: 1939: 1937: 1934: 1932: 1929: 1927: 1924: 1922: 1919: 1917: 1914: 1912: 1909: 1906: 1905: 1899: 1898: 1896: 1894: 1890: 1882: 1879: 1877: 1874: 1872: 1869: 1867: 1864: 1863: 1862: 1859: 1855: 1852: 1851: 1850: 1847: 1845: 1844: 1839: 1837: 1836:Matrix normal 1834: 1832: 1829: 1826: 1825: 1820: 1816: 1813: 1812: 1811: 1808: 1806: 1805: 1802:Multivariate 1800: 1798: 1795: 1793: 1790: 1788: 1785: 1781: 1778: 1777: 1776: 1773: 1770: 1766: 1762: 1759: 1757: 1754: 1753: 1752: 1749: 1747: 1744: 1741: 1737: 1736: 1734: 1732: 1729:Multivariate 1726: 1716: 1713: 1712: 1710: 1704: 1701: 1695: 1685: 1682: 1680: 1677: 1675: 1673: 1669: 1667: 1665: 1661: 1659: 1657: 1653: 1651: 1649: 1644: 1642: 1640: 1635: 1633: 1631: 1626: 1624: 1622: 1617: 1615: 1613: 1608: 1606: 1603: 1601: 1598: 1596: 1593: 1591: 1588: 1587: 1585: 1581:with support 1579: 1573: 1570: 1568: 1565: 1563: 1560: 1558: 1557: 1552: 1550: 1547: 1545: 1542: 1540: 1537: 1535: 1532: 1530: 1527: 1525: 1524: 1519: 1517: 1514: 1510: 1507: 1506: 1505: 1502: 1500: 1497: 1495: 1494: 1486: 1484: 1481: 1479: 1476: 1474: 1471: 1469: 1466: 1464: 1461: 1459: 1456: 1454: 1453: 1448: 1446: 1443: 1441: 1440: 1435: 1433: 1430: 1428: 1425: 1424: 1422: 1418:on the whole 1414: 1408: 1405: 1401: 1398: 1397: 1396: 1393: 1391: 1390:type-2 Gumbel 1388: 1386: 1383: 1381: 1378: 1376: 1373: 1371: 1368: 1366: 1363: 1361: 1358: 1356: 1353: 1351: 1348: 1346: 1343: 1341: 1338: 1336: 1333: 1331: 1328: 1326: 1323: 1321: 1318: 1316: 1313: 1311: 1308: 1306: 1303: 1301: 1298: 1296: 1293: 1291: 1288: 1284: 1281: 1280: 1279: 1276: 1274: 1272: 1267: 1265: 1262: 1260: 1259:Half-logistic 1257: 1253: 1250: 1249: 1248: 1245: 1243: 1240: 1236: 1233: 1231: 1228: 1227: 1226: 1223: 1221: 1218: 1216: 1215:Folded normal 1213: 1209: 1206: 1205: 1204: 1203: 1199: 1195: 1192: 1190: 1187: 1185: 1182: 1181: 1180: 1177: 1173: 1170: 1169: 1168: 1165: 1163: 1160: 1158: 1155: 1149: 1146: 1145: 1144: 1141: 1139: 1136: 1135: 1134: 1131: 1129: 1126: 1124: 1121: 1119: 1116: 1114: 1111: 1109: 1106: 1104: 1101: 1100: 1098: 1090: 1084: 1081: 1079: 1076: 1074: 1071: 1069: 1066: 1064: 1061: 1059: 1058:Raised cosine 1056: 1054: 1051: 1049: 1046: 1044: 1041: 1039: 1036: 1034: 1031: 1029: 1026: 1024: 1021: 1019: 1016: 1014: 1011: 1009: 1006: 1004: 1001: 999: 996: 995: 993: 987: 984: 978: 968: 965: 963: 960: 958: 955: 953: 950: 948: 945: 943: 940: 938: 935: 933: 932:Mixed Poisson 930: 928: 925: 923: 920: 918: 915: 913: 910: 908: 905: 903: 900: 898: 895: 893: 890: 888: 885: 883: 880: 879: 877: 871: 865: 862: 860: 857: 855: 852: 850: 847: 845: 842: 840: 837: 833: 830: 829: 828: 825: 823: 820: 818: 815: 813: 812:Beta-binomial 810: 808: 805: 803: 800: 799: 797: 791: 788: 782: 777: 773: 766: 761: 759: 754: 752: 747: 746: 743: 736: 735:0-471-12844-9 732: 728: 724: 721: 718: 714: 711: 707: 703: 700: 696: 695: 687: 684: 682: 679: 677: 674: 672: 669: 668: 662: 660: 655: 653: 649: 645: 641: 637: 633: 628: 626: 622: 618: 614: 610: 606: 596: 594: 590: 587: 583: 575: 569: 566: 565:genetic drift 562: 558: 555: 551: 550: 549: 531: 528: 525: 520: 516: 512: 507: 502: 499: 496: 492: 488: 483: 479: 475: 472: 469: 466: 461: 457: 453: 450: 445: 441: 437: 434: 429: 425: 417: 416: 415: 399: 395: 391: 388: 385: 380: 376: 367: 363: 344: 338: 333: 329: 321: 317: 312: 303: 299: 294: 286: 281: 278: 275: 271: 261: 258: 255: 252: 249: 243: 237: 234: 231: 225: 220: 217: 211: 205: 202: 197: 193: 189: 186: 183: 178: 174: 167: 164: 157: 156: 155: 150: 146: 140: 136: 132: 128: 124: 121: 117: 107: 105: 101: 97: 96:probabilities 93: 89: 78: 75: 67: 57: 53: 47: 46: 40: 36: 32: 27: 18: 17: 2017: 2005: 1971:Multivariate 1970: 1958: 1946: 1941:Wrapped LĂ©vy 1901: 1849:Matrix gamma 1842: 1822: 1810:Normal-gamma 1803: 1769:Continuous: 1768: 1745: 1739: 1684:Tukey lambda 1671: 1663: 1658:-exponential 1655: 1647: 1638: 1629: 1620: 1614:-exponential 1611: 1555: 1522: 1489: 1451: 1438: 1365:Poly-Weibull 1310:Log-logistic 1270: 1269:Hotelling's 1201: 1043:Logit-normal 917:Gauss–Kuzmin 912:Flory–Schulz 793:with finite 726: 716: 709: 698: 656: 651: 647: 643: 639: 635: 631: 629: 624: 620: 612: 608: 604: 602: 592: 588: 579: 553: 547: 361: 359: 148: 138: 119: 116:Warren Ewens 113: 91: 85: 70: 61: 50:Please help 42: 2055:Exponential 1904:directional 1893:Directional 1780:Generalized 1751:Multinomial 1706:continuous- 1646:Kaniadakis 1637:Kaniadakis 1628:Kaniadakis 1619:Kaniadakis 1610:Kaniadakis 1562:Tracy–Widom 1539:Skew normal 1521:Noncentral 1305:Log-Laplace 1283:Generalized 1264:Half-normal 1230:Generalized 1194:Logarithmic 1179:Exponential 1133:Chi-squared 1073:U-quadratic 1038:Kumaraswamy 980:Continuous 927:Logarithmic 822:Categorical 368:, whenever 135:probability 64:August 2011 56:introducing 2133:Categories 2050:Elliptical 2006:Degenerate 1992:Degenerate 1740:Discrete: 1699:univariate 1554:Student's 1509:Asymmetric 1488:Johnson's 1416:supported 1360:Phase-type 1315:Log-normal 1300:Log-Cauchy 1290:Kolmogorov 1208:Noncentral 1138:Noncentral 1118:Beta prime 1068:Triangular 1063:Reciprocal 1033:Irwin–Hall 982:univariate 962:Yule–Simon 844:Rademacher 786:univariate 634:is taken, 580:This is a 572:See also: 110:Definition 1775:Dirichlet 1756:Dirichlet 1666:-Gaussian 1641:-Logistic 1478:Holtsmark 1450:Gaussian 1437:Fisher's 1420:real line 922:Geometric 902:Delaporte 807:Bernoulli 784:Discrete 729:, Wiley. 493:∑ 470:⋯ 389:… 295:θ 272:∏ 259:− 250:θ 244:⋯ 232:θ 226:θ 206:θ 187:… 168:⁡ 133:then the 2108:Category 2040:Circular 2033:Families 2018:Singular 1997:singular 1761:Negative 1708:discrete 1674:-Weibull 1632:-Weibull 1516:Logistic 1400:Discrete 1370:Rayleigh 1350:Nakagami 1273:-squared 1247:Gompertz 1096:interval 832:Negative 817:Binomial 665:See also 561:mutation 2118:Commons 2090:Wrapped 2085:Tweedie 2080:Pearson 2075:Mixture 1982:Bingham 1881:Complex 1871:Inverse 1861:Wishart 1854:Inverse 1841:Matrix 1815:Inverse 1731:(joint) 1650:-Erlang 1504:Laplace 1395:Weibull 1252:Shifted 1235:Inverse 1220:FrĂ©chet 1143:Inverse 1078:Uniform 998:Arcsine 957:Skellam 952:Poisson 875:support 849:Soliton 802:Benford 795:support 638:of the 145:alleles 123:gametes 100:alleles 52:improve 2024:Cantor 1866:Normal 1697:Mixed 1623:-Gamma 1549:Stable 1499:Landau 1473:Gumbel 1427:Cauchy 1355:Pareto 1167:Erlang 1148:Scaled 1103:Benini 942:Panjer 733:  104:sample 1746:Ewens 1572:Voigt 1544:Slash 1325:Lomax 1320:Log-t 1225:Gamma 1172:Hyper 1162:Davis 1157:Dagum 1013:Bates 1003:ARGUS 887:Borel 692:Notes 619:. As 603:When 131:locus 37:, or 1995:and 1953:Kent 1380:Rice 1295:LĂ©vy 1123:Burr 1053:PERT 1018:Beta 967:Zeta 859:Zipf 776:list 731:ISBN 563:and 127:gene 1831:LKJ 1128:Chi 708:", 86:In 2135:: 661:. 654:. 595:. 165:Pr 106:. 90:, 41:, 33:, 1843:t 1804:t 1672:q 1664:q 1656:q 1648:Îș 1639:Îș 1630:Îș 1621:Îș 1612:Îș 1556:t 1523:t 1492:U 1490:S 1452:q 1439:z 1271:T 1202:F 778:) 774:( 764:e 757:t 750:v 737:. 652:n 648:m 644:m 640:n 636:m 632:n 625:n 621:Ξ 613:Ξ 609:n 605:Ξ 589:n 554:n 532:. 529:n 526:= 521:i 517:a 513:i 508:n 503:1 500:= 497:i 489:= 484:n 480:a 476:n 473:+ 467:+ 462:3 458:a 454:3 451:+ 446:2 442:a 438:2 435:+ 430:1 426:a 400:n 396:a 392:, 386:, 381:1 377:a 362:Ξ 345:, 339:! 334:j 330:a 322:j 318:a 313:j 304:j 300:a 287:n 282:1 279:= 276:j 265:) 262:1 256:n 253:+ 247:( 241:) 238:1 235:+ 229:( 221:! 218:n 212:= 209:) 203:; 198:n 194:a 190:, 184:, 179:1 175:a 171:( 152:2 149:a 142:1 139:a 120:n 77:) 71:( 66:) 62:( 48:.

Index

list of references
related reading
external links
inline citations
improve
introducing
Learn how and when to remove this message
population genetics
probabilities
alleles
sample
Warren Ewens
gametes
gene
locus
probability
alleles
population mutation rate
mutation
genetic drift
Infinite-alleles model
probability distribution
partitions of the integer
random permutation
Chinese restaurant process
Chinese restaurant table distribution
Coalescent theory
Unified neutral theory of biodiversity
Biomathematics
The Ubiquitous Ewens Sampling Formula

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