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22:
1896:
1606:
Number of permutations of 3 distinct letters (ABC) each with n copies such that one (1) fixed points. E.g. if AAAAABBBBBCCCCC n=3*5 letters permutations then one fixed points n5=15150 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 02 2006
609:
1, 0, 0, 1, 1, 0, 4, 0, 1, 10, 24, 27, 16, 12, 0, 1, 297, 672, 736, 480, 246, 64, 24, 0, 1, 13756, 30480, 32365, 21760, 10300, 3568, 970, 160, 40, 0, 1, 925705, 2016480, 2116836, 1418720, 677655, 243360, 67920, 14688, 2655, 320, 60, 0, 1
2065:
1063:
COMMENT Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears twice. If there is only one letter of each type we get A000166. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 15
1545:
Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears thrice. If there is only one letter of each type we get A000166. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 15
2202:
229:
1818:
140:
2414:
2317:
1953:
2604:
219:
2594:
1056:
sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k);seq(f(0, n, 2)/2!^n, n=0..18); (AUTHOR Barbara Haas
Margolius (margolius(AT)math.csuohio.edu) )
130:
1539:
sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); seq(f(0, n, 3)/3!^n, n=0..18); (AUTHOR Barbara Haas
Margolius (margolius(AT)math.csuohio.edu)
1968:
2589:
2609:
2599:
2269:
On the article page the first table is not very pretty. There is no horizontal bar under number 2,3,4,5,6,7 and the verticle bar goes down too far:
106:
195:
2565:
2107:
97:
58:
186:
163:
2249:
My version doesn't require knowing how many permutations a set has, given its cardinality. In that sense it is simpler.
1891:{\displaystyle X_{i}={\begin{cases}1&{\text{if }}i{\text{ is a fixed point}},\\0&{\text{otherwise}}.\end{cases}}}
33:
1493:
00166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.]
1008:
00166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.]
2375:
2278:
2569:
21:
2254:
2075:
1907:
253:
A008290 Triangle T(n,k) of rencontres numbers (number of *permutations of n elements with k fixed points).
1681:
Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal)
39:
83:
178:
157:
2561:
2544:
1840:
2548:
194:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
2250:
2071:
1753:
1740:
89:
600:
A059056 Penrice
Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers).
73:
52:
2516:
2496:
2236:
1803:
1047:
A059072 Penrice
Christmas gift numbers; card-matching numbers; dinner-diner matching numbers.
606:
COMMENT: Analogous to A008290. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 10 2005
2226:
For any wikipedians wondering, I should point out that the work or particularly the results
1960:
2229:
are not original work. The results have probably been known for a couple of centuries.
2213:
Here the probability distribution is discrete, so taking the expectation is a sumation
1705:
1699:
1693:
2583:
2512:
2492:
2232:
1799:
1724:
charcters:quadruple, example:AAAA, AAAABBBB, AAAABBBBCCCC, AAAABBBBCCCCDDDD, etc...
2060:{\displaystyle E(X_{1})+\cdots +E(X_{n})={\frac {1}{n}}+\cdots +{\frac {1}{n}}=1.}
1680:
102:
191:
79:
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2552:
2520:
2506:
I've now just noticed that the tables look fine using Opera and IE browsers.
2500:
2258:
2240:
2079:
1807:
2541:
1730:
charcters:quintuple, example:AAAAA, AAAAABBBBB, AAAAABBBBBCCCCC, etc...
2367:
Perhaps one solution would be to put in blanks in those extra cells.
2542:
http://en.wikipedia.org/Formal_power_series#Extracting_coefficients
2509:
The only problem is when they are viewed in
Firefox (version 3.5.1)
2099:(I'm now going to make the effort to write the equations properly)
1712:
2197:{\displaystyle E(\sum _{i=1}^{n}X_{i})=\sum _{i=1}^{n}E(X_{i})}
1532:
A059073 Card-matching numbers (Dinner-Diner matching numbers).
15:
1782:
is E_n = sum_p_from_1_to_n! { sum_m_from_1_to_n { X } } / n!
1516:
AAABBBCCCDDD (table sign: 3333)then 13833 derangements, etc.
2558:
Replace approximation by lim specifying range of validity.
1776:
Now let X=1 if in the p_th permutation, element m is fixed,
1884:
2092:
I'd argue that the two proofs are almost almost identical.
1785:
E_n = sum_m_from_1_to_n { sum_p_from_1_to_n! { X } } / n!
1576:
etc... A000172 Franel number a(n) = Sum C(n,k)^3, k=0..n.
1537:
k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)-: -->
1054:
k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)-: -->
249:
Extension: If all character once : example: ABCDE......
1527:
1, 0, 1, 56, 13833, 6699824, 5691917785, 7785547001784,
1488:
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961...
1031:
AABBCCDD (table sign: 2222)then 297 derangements, etc.
1003:
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961...
2378:
2281:
2110:
1971:
1910:
1821:
190:, a collaborative effort to improve the coverage of
101:, a collaborative effort to improve the coverage of
2490:
It is not ideal, but I think it looks a bit better.
2095:The same arguement presented slightly differently.
1513:
AAABBBCCC (table sign: 333)then 56 derangements,
2408:
2311:
2196:
2059:
1947:
1890:
1477:"ABCD" (table sign: 1111)then 9 derangements, etc.
992:"ABCD" (table sign: 1111)then 9 derangements, etc.
2088:Well, I certainly prefer your equation mark-up.
1711:...mirror of central and multiply of diagonal...
1510:AAABBB (table sign: 33)then 1 derangements,
1028:AABBCC (table sign: 222)then 10 derangements,
2398:
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1733:table 1.column :5, 55, 555, 5555, 55555, etc...
1727:table 1.column :4, 44, 444, 4444, 44444, etc...
1714:(Pascal háromszög tükrözése és szorzás. Minta.)
1504:"0" (table sign: "0")then 1 derangements,
1019:"0" (table sign: "0")then 1 derangements,
2223:In my version, I've just shown that explicitly
2220:, we just need to swap the order of summation.
1671:where is it :formula or generating function(?)
1507:AAA (table sign: 3)then 0 derangements,
1087:where is it :formula or generating function(?)
1025:AABB (table sign: 22)then 1 derangements,
1736:a great number of connexion of interesting !!
1042:1, 0, 1, 10, 297, 13756, 925705, 85394646,...
1022:AA (table sign: 2)then 0 derangements,
8:
1772:, the expected number of fixed points is 1 :
1474:"ABC" (table sign: 111)then 2 derangements,
989:"ABC" (table sign: 111)then 2 derangements,
597:If all character twice: example: AABBCC....
19:
2409:{\displaystyle _{n}\!\!\diagdown \!\!^{k}}
2312:{\displaystyle _{n}\!\!\diagdown \!\!^{k}}
152:
47:
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1774:We'll number the permutations p = 1 to n!
1471:"AB" (table sign: 11)then 1 derangements,
1465:"0" (table sign: "0")then 1 derangements,
986:"AB" (table sign: 11)then 1 derangements,
980:"0" (table sign: "0")then 1 derangements,
2369:
2272:
1788:E_n = sum_m_from_1_to_n { (n-1)! } / n!
1780:Now the expected number of fixed points
1461:If original or classic table: (1.table)
1100:
976:If original or classic table: (1.table)
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1943:
1468:"A" (table sign: 1)then 0 derangements,
983:"A" (table sign: 1)then 0 derangements,
154:
49:
1948:{\displaystyle X_{1}+\cdots +X_{n},\,}
1104:fixed point: character numbers:
621:fixed point: character numbers:
267:fixed point: character numbers:
1536:FORMULA: MAPLE p := (x, k)-: -->
7:
1053:FORMULA: MAPLE p := (x, k)-: -->
184:This article is within the scope of
95:This article is within the scope of
1902:Then the number of fixed points is
1760:Justification that expectation is 1
38:It is of interest to the following
2605:Low-importance Statistics articles
1554:2.column (free or "0" -fixed point
14:
2595:Low-priority mathematics articles
115:Knowledge:WikiProject Mathematics
2590:Start-Class mathematics articles
204:Knowledge:WikiProject Statistics
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118:Template:WikiProject Mathematics
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2600:Start-Class Statistics articles
1655:etc... A000489 Card matching.
1630:etc... A000535 Card matching.
1538:p(x, k)^n; f := (t, n, k)-: -->
1055:p(x, k)^n; f := (t, n, k)-: -->
224:This article has been rated as
207:Template:WikiProject Statistics
135:This article has been rated as
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2016:
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1756:28. jun. 2007. 16. apr. 2009.
1601:etc... A000279 Card matching.
1:
2540:. Definition can be found at
198:and see a list of open tasks.
109:and see a list of open tasks.
2532:Explain meaning of symbol z.
1636:5.column ( "3" fixed point)
1611:4.column ( "2" fixed point)
1582:3.column ( "1" -fixed point)
2553:08:37, 10 August 2013 (UTC)
2527:Formulae need clarification
2626:
1963:number of fixed points is
1676:where is it :bibliography?
1092:where is it :bibliography?
246:START Zlajos 17 jun 2007
2574:22:18, 5 March 2012 (UTC)
2521:23:54, 19 July 2009 (UTC)
2501:23:23, 19 July 2009 (UTC)
2259:01:07, 20 July 2009 (UTC)
2241:20:52, 19 July 2009 (UTC)
2080:15:01, 19 July 2009 (UTC)
1808:08:42, 19 July 2009 (UTC)
1778:when it is not fixed, X=0
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2102:You use the fact that
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1813:Simpler argument: Let
1791:E_n = n * (n-1)! / n!
187:WikiProject Statistics
28:This article is rated
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1798:(with thanks to FD)
121:mathematics articles
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1501:analogous (3.table)
1082:column: 2,3,4,5,...
1016:analogous (2.table)
210:Statistics articles
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1603:] COMMENT
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226:Low-importance
222:
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196:the discussion
182:
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167:Low‑importance
161:
149:
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127:
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107:the discussion
94:
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77:
65:
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56:
44:
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26:
13:
10:
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2256:
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2251:Michael Hardy
2248:
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2238:
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2222:
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2072:Michael Hardy
2070:
2054:
2051:
2046:
2043:
2038:
2035:
2032:
2027:
2024:
2019:
2011:
2007:
2000:
1997:
1994:
1991:
1983:
1979:
1972:
1965:
1964:
1962:
1958:
1940:
1935:
1931:
1927:
1924:
1921:
1916:
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1708:
1706:
1703:A129536 : -->
1702:
1700:
1697:A129352 : -->
1696:
1694:
1691:A113899 : -->
1690:
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1646:333 :435
1644:
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1621:333 :378
1619:
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1592:333 :216
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1570:444 :346
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1007:
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951:
948:
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929:
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772:
769:
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750:
748:
745:
742:
739:
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730:
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726:
724:
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718:
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714:
712:
710:
708:
705:
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699:
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629:
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623:
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611:
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588:
586:
584:
582:
579:
576:
573:
570:
567:
564:
561:
558:
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551:
549:
547:
545:
543:
541:
538:
535:
532:
529:
526:
523:
520:
517:
516:
513:
511:
509:
507:
505:
503:
501:
498:
495:
492:
489:
486:
483:
480:
479:
476:
474:
472:
470:
468:
466:
464:
462:
459:
456:
453:
450:
447:
444:
443:
440:
438:
436:
434:
432:
430:
428:
426:
424:
421:
418:
415:
412:
409:
408:
405:
403:
401:
399:
397:
395:
393:
391:
389:
387:
384:
381:
378:
375:
374:
371:
369:
367:
365:
363:
361:
359:
357:
355:
353:
351:
348:
345:
342:
341:
338:
336:
334:
332:
330:
328:
326:
324:
322:
320:
318:
316:
313:
310:
309:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
265:
259:
254:
252:
251:
250:
247:
231:
227:
221:
218:
217:
214:
197:
193:
189:
188:
183:
180:
176:
175:
171:
165:
162:
159:
155:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2560:— Preceding
2537:
2489:
2366:
2268:
2265:Table format
2217:
2216:So to prove
2206:
1797:
1769:
1765:
1763:
1752:
1739:
1654:
1651:
1648:
1645:
1643:222 :16
1642:
1639:
1629:
1626:
1623:
1620:
1618:222 :27
1617:
1614:
1605:
1600:
1597:
1594:
1591:
1589:222 :24
1588:
1585:
1575:
1572:
1569:
1567:333 :56
1566:
1564:222 :10
1563:
1560:
1557:
1549:
1535:
1521:column : -->
1496:
1482:column : -->
1460:
1400:14125889160
1397:24571261710
1394:32659846104
1391:31234760055
1388:19180338840
1107:free or "0"
1067:
1058:
1052:
1036:column : -->
1011:
997:column : -->
975:
624:free or "0"
608:
605:
270:free or "0"
248:
245:
225:
185:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
1640:111 :1
1615:111 :0
1586:111 :3
1561:111 :2
1522:free or 0 :
1483:free or 0 :
1406:2375679240
1403:6433608330
1385:5691917785
1037:free or 0 :
998:free or 0 :
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2584:Categories
1719:continued:
1412:182701480
1409:722303568
943:128058000
940:191384599
937:183749160
201:Statistics
192:statistics
164:Statistics
2545:Heycarnut
1875:otherwise
1709:Demo: -->
1415:38712600
1356:17399178
1353:30573900
1350:40563765
1347:38358540
1344:23123880
1072:Question:
949:22558928
946:61585776
934:85394646
2562:unsigned
1961:expected
1426:3333333
1421:1035330
1418:6889320
1362:2729295
1359:7723640
1341:6699824
955:1507392
952:6506955
931:2222222
899:1418720
896:2116836
893:2016480
556:1111111
2513:Pnelnik
2493:Pnelnik
2233:Pnelnik
1959:so the
1800:Pnelnik
1794:E_n = 1
1661:3.table
1382:333333
1368:180100
1365:776520
1097:3.table
1077:2.table
958:284550
905:243360
902:677655
890:925705
887:222222
614:2.table
518:111111
260:1.table
228:on the
139:on the
2207:(eqn1)
1793:=: -->
1790:=: -->
1787:=: -->
1784:=: -->
1754:Zlajos
1741:Zlajos
1497:then:
1371:33372
1338:33333
1315:16476
1312:38448
1309:69039
1306:90944
1303:84510
1300:49464
1297:13833
1012:then:
961:43848
911:14688
908:67920
860:10300
857:21760
854:32365
851:30480
848:13756
845:22222
481:11111
36:scale.
1747:copy:
1710:: -->
1704:: -->
1698:: -->
1692:: -->
1374:5355
1321:1431
1318:5184
1294:3333
964:5901
914:2655
863:3568
805:2222
562:1855
559:1854
445:1111
2570:talk
2549:talk
2517:talk
2497:talk
2255:talk
2237:talk
2218:eqn1
2076:talk
1804:talk
1686:OEIS
1546:2006
1435:etc
1377:540
1324:216
1271:189
1268:324
1265:435
1262:378
1259:216
1253:333
1148:"0"
1064:2006
967:560
917:320
869:160
866:970
820:246
817:480
814:736
811:672
808:297
767:222
665:"0"
568:315
565:924
527:135
524:264
521:265
410:111
311:"0"
1850:if
1327:54
1277:27
1274:54
1256:56
1215:33
1143:12
1140:11
1137:10
970:84
920:60
872:40
826:24
823:64
782:12
779:16
776:27
773:24
770:10
731:22
660:12
657:11
654:10
574:21
571:70
533:15
530:40
493:10
490:20
487:45
484:44
376:11
306:12
303:11
300:10
220:Low
131:Low
2586::
2572:)
2551:)
2519:)
2499:)
2472:1
2469:0
2466:1
2447:1
2444:0
2439:7
2393:╲
2361:1
2358:0
2355:1
2350:1
2347:0
2342:7
2296:╲
2257:)
2239:)
2156:∑
2119:∑
2078:)
2055:1.
2036:⋯
1995:⋯
1925:⋯
1806:)
1768:≥
1657:]
1632:]
1578:]
1541:]
1333:1
1330:0
1283:1
1280:0
1236:1
1233:0
1230:9
1227:0
1224:9
1221:0
1218:1
1192:1
1189:0
1186:0
1183:0
1180:3
1151:1
1134:9
1131:8
1128:7
1125:6
1122:5
1119:4
1116:3
1113:2
1110:1
926:1
923:0
878:1
875:0
832:1
829:0
788:1
785:0
746:1
743:0
740:4
737:0
734:1
706:1
703:0
700:0
697:2
668:1
651:9
648:8
645:7
642:6
639:5
636:4
633:3
630:2
627:1
580:1
577:0
539:1
536:0
499:1
496:0
460:1
457:0
454:6
451:8
448:9
422:1
419:0
416:3
413:2
385:1
382:0
379:1
349:1
346:0
343:1
314:1
297:9
294:8
291:7
288:6
285:5
282:4
279:3
276:2
273:1
2568:(
2547:(
2515:(
2495:(
2436:6
2433:5
2430:4
2427:3
2424:2
2421:1
2418:0
2402:k
2384:n
2339:6
2336:5
2333:4
2330:3
2327:2
2324:1
2321:0
2305:k
2287:n
2253:(
2235:(
2192:)
2187:i
2183:X
2179:(
2176:E
2171:n
2166:1
2163:=
2160:i
2152:=
2149:)
2144:i
2140:X
2134:n
2129:1
2126:=
2123:i
2115:(
2112:E
2074:(
2052:=
2047:n
2044:1
2039:+
2033:+
2028:n
2025:1
2020:=
2017:)
2012:n
2008:X
2004:(
2001:E
1998:+
1992:+
1989:)
1984:1
1980:X
1976:(
1973:E
1941:,
1936:n
1932:X
1928:+
1922:+
1917:1
1913:X
1879:.
1869:0
1862:,
1854:i
1844:1
1838:{
1833:=
1828:i
1824:X
1802:(
1770:1
1766:n
1749:]
1049:]
602:]
255:]
232:.
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
