228:
31:
379:
372:
365:
358:
351:
344:
337:
330:
324:
1722:
168:
people subscribe to a telephone service that can connect any two of them by a call, but cannot make a single call connecting more than two people. How many different patterns of connection are possible? For instance, with three subscribers, there are three ways of forming a single telephone call, and
184:
of the people that is its own inverse. In this permutation, each two people who call each other are swapped, and the people not involved in calls remain fixed in place. Conversely, every possible involution has the form of a set of pairwise swaps of this type. Therefore, the telephone numbers also
959:
1412:
1470:
1267:
263:. Inverting a permutation corresponds to swapping the two tableaux, and so the self-inverse permutations correspond to single tableaux, paired with themselves. Thus, the telephone numbers also count the number of Young tableaux with
754:
279:
of permutations, and the Young tableaux with a given shape form a basis of the irreducible representation with that shape. Therefore, the telephone numbers give the sum of the degrees of the irreducible representations.
683:
2207:
Solomon, A. I.; Blasiak, P.; Duchamp, G.; Horzela, A.; Penson, K.A. (2005), "Combinatorial physics, normal order and model
Feynman graphs", in Gruber, Bruno J.; Marmo, Giuseppe; Yoshinaga, Naotaka (eds.),
1914:
169:
one additional pattern in which no calls are being made, for a total of four patterns. For this reason, the numbers counting how many patterns are possible are sometimes called the telephone numbers.
1138:
1283:
1792:
1158:
704:
subscribers to a telephone system into the patterns in which the first person is not calling anyone else, and the patterns in which the first person is making a call. There are
1028:
568:
715:
connection patterns in which the first person is disconnected, explaining the first term of the recurrence. If the first person is connected to someone, there are
1051:
1717:{\displaystyle \sum _{n}T(n){\frac {x^{n-1}}{(n-1)!}}=\sum _{n}T(n-1){\frac {x^{n-1}}{(n-1)!}}+x\sum _{n}T(n-2){\frac {x^{n-2}}{(n-2)!}}\,\mathrm {,~which~is} }
979:
516:
mutually non-attacking rooks, where two configurations are counted as essentially different if there is no symmetry of the board that takes one into the other.
572:
1438:-th derivative of this function. The exponential generating function can be derived in a number of ways; for example, taking the recurrence relation for
2723:
2038:
150:
1842:
2805:
256:
255:
into these squares in such a way that the numbers increase from left to right and from top to bottom throughout the tableau. According to the
954:{\displaystyle T(n)=\sum _{k=0}^{\lfloor n/2\rfloor }{\binom {n}{2k}}(2k-1)!!=\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n!}{2^{k}(n-2k)!k!}}.}
2350:
Beissinger, Janet
Simpson (1987), "Similar constructions for Young tableaux and involutions, and their application to shiftable tableaux",
1059:
2800:
2235:
247:
with a horizontal top edge, a vertical left edge, and a single monotonic chain of edges from top right to bottom left. A standard
2810:
2315:
2348:
A direct bijection between involutions and tableaux, inspired by the recurrence relation for the telephone numbers, is given by
512:, these numbers form one of the key components of a formula for the overall number of "essentially different" configurations of
2661:
2556:
2500:
2080:
1277:
2607:
2352:
1839:-th telephone number, and the telephone numbers can also be realized as certain special values of the Hermite polynomials:
1726:
2103:
509:
508:), and in such a way that the configuration of the rooks is symmetric under a diagonal reflection of the board. Via the
161:
1148:
2795:
2598:
484:
272:
118:
186:
1140:
is the product of the odd integers up to its argument and counts the number of ways of completely matching the
177:
102:
2659:
Kim, Dongsu; Kim, Jang Soo (2010), "A combinatorial approach to the power of 2 in the number of involutions",
535:
197:
78:
216:, for which any pattern of pairwise connections is possible; thus, the Hosoya index of a complete graph on
2495:
1407:{\displaystyle G(x)=\sum _{n=0}^{\infty }{\frac {T(n)x^{n}}{n!}}=\exp \left(x+{\frac {x^{2}}{2}}\right).}
2815:
686:
268:
201:
126:
987:
77:
people can be connected by person-to-person telephone calls. These numbers also describe the number of
2259:
2281:
2137:
982:
2605:; Gardy, Danièle; Gouyou-Beauchamps, Dominique (2002), "Generating functions for generating trees",
227:
1944:
1828:
1152:
530:
505:
130:
2773:
2747:
2696:
2670:
2642:
2616:
2527:
2467:
2459:
2425:
2397:
2271:
2241:
2213:
2179:
1824:
689:, by which they may easily be calculated. One way to explain this recurrence is to partition the
106:
1262:{\displaystyle T(n)\sim \left({\frac {n}{e}}\right)^{n/2}{\frac {e^{\sqrt {n}}}{(4e)^{1/4}}}\,.}
2602:
2491:
2332:
2231:
189:
problem studied by Rothe in 1800 and these numbers have also been called involution numbers.
2757:
2680:
2626:
2565:
2509:
2451:
2407:
2361:
2324:
2223:
2171:
1054:
70:
24:
2769:
2692:
2638:
2579:
2523:
2421:
2375:
2307:
2293:
2191:
2149:
2090:
2765:
2688:
2634:
2575:
2519:
2417:
2371:
2289:
2187:
2145:
2086:
2025:
276:
236:
122:
2285:
2141:
2122:
1033:
964:
213:
86:
35:
2630:
2789:
2777:
2531:
2471:
2366:
1415:
260:
248:
196:, a subset of the edges of a graph that touches each vertex at most once is called a
110:
2700:
2646:
2429:
2245:
2075:
1940:
1932:
205:
193:
82:
1414:
In other words, the telephone numbers may be read off as the coefficients of the
2713:
181:
94:
58:
2684:
1823:
This function is closely related to the exponential generating function of the
23:
This article is about the integer sequence. For the phone network address, see
2761:
2738:
Amdeberhan, Tewodros; Moll, Victor (2015), "Involutions and their progenies",
2412:
1831:
of the complete graphs. The sum of absolute values of the coefficients of the
501:
478:
A diagonally symmetric non-attacking placement of eight rooks on a chessboard
2328:
2227:
2544:
2258:
Blasiak, P.; Dattoli, G.; Horzela, A.; Penson, K. A.; Zhukovsky, K. (2008),
749:
244:
2570:
2514:
2498:; Moore, W. K. (1951), "On recursions connected with symmetric groups. I",
2388:
Halverson, Tom; Reeks, Mike (2015), "Gelfand models for diagram algebras",
2336:
2020:
by computing the recurrence for the sequence of telephone numbers, modulo
30:
2218:
2162:
Getu, Seyoum (1991), "Evaluating determinants via generating functions",
2260:"Motzkin numbers, central trinomial coefficients and hybrid polynomials"
185:
count involutions. The problem of counting involutions was the original
2463:
2183:
2308:"Extremal problems for topological indices in combinatorial chemistry"
204:, where the graphs model molecules and the number of matchings is the
2621:
678:{\displaystyle T(n)=T(n-1)+(n-1)T(n-2),\quad \mathrm {for~} n\geq 1,}
259:, permutations correspond one-for-one with ordered pairs of standard
2455:
2175:
2016:, one can test whether there exists a telephone number divisible by
164:
provides the following explanation for these numbers: suppose that
2752:
2675:
2402:
2276:
226:
133:
by which they may be calculated, giving the values (starting from
29:
504:
in such a way that no two rooks attack each other (the so-called
1147:
selected elements. It follows from the summation formula and
1909:{\displaystyle T(n)={\frac {{\mathit {He}}_{n}(i)}{i^{n}}}.}
1870:
1867:
143:
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence
2717:
2033:
2028:. The primes that divide at least one telephone number are
145:
2048:. Each of them divides infinitely many telephone numbers.
487:, the telephone numbers count the number of ways to place
200:. Counting the matchings of a given graph is important in
2123:"Generating functions via Hankel and Stieltjes matrices"
1794:
The general solution to this differential equation is
992:
740:
people, explaining the second term of the recurrence.
1845:
1729:
1473:
1286:
1161:
1062:
1036:
990:
967:
757:
575:
538:
2031:
2, 5, 13, 19, 23, 29, 31, 43, 53, 59, ... (sequence
748:
The telephone numbers may be expressed exactly as a
105:, the sum of absolute values of coefficients of the
1133:{\displaystyle (2k-1)!!={\frac {(2k)!}{2^{k}\,k!}}}
125:. Involution numbers were first studied in 1800 by
2085:, Reading, Mass.: Addison-Wesley, pp. 65–67,
1908:
1786:
1716:
1406:
1261:
1132:
1045:
1022:
973:
953:
677:
562:
2044:The odd primes in this sequence have been called
1820:shows that the constant of proportionality is 1.
1434:-th telephone number is the value at zero of the
829:
811:
2212:, Kluwer Academic Publishers, pp. 527–536,
284:
239:is a geometric shape formed by a collection of
1835:-th (probabilist's) Hermite polynomial is the
172:Every pattern of pairwise connections between
46:has ten matchings, corresponding to the value
1931:-th telephone number is divisible by a large
1013:
995:
8:
892:
878:
803:
789:
251:is formed by placing the numbers from 1 to
2306:Tichy, Robert F.; Wagner, Stephan (2005),
1030:counts the number of ways of choosing the
208:. The largest possible Hosoya index of an
2751:
2724:On-Line Encyclopedia of Integer Sequences
2674:
2620:
2569:
2513:
2411:
2401:
2365:
2275:
2217:
1895:
1875:
1866:
1865:
1861:
1844:
1780:
1728:
1682:
1681:
1647:
1641:
1617:
1573:
1567:
1543:
1502:
1496:
1478:
1472:
1385:
1379:
1342:
1323:
1317:
1306:
1285:
1255:
1242:
1238:
1216:
1210:
1200:
1196:
1182:
1160:
1120:
1114:
1090:
1061:
1035:
1012:
994:
991:
989:
966:
912:
897:
884:
877:
866:
828:
810:
808:
795:
788:
777:
756:
733:patterns of connection for the remaining
649:
574:
537:
271:, the Ferrers diagrams correspond to the
117:cells, and the sum of the degrees of the
2592:
2590:
2588:
2442:Holt, D. F. (1974), "Rooks inviolate",
2202:
2200:
2070:
2068:
2066:
2064:
2062:
2060:
2056:
981:gives the number of matched pairs, the
2547:; Wyman, Max (1955), "On solutions of
2486:
2484:
2482:
2480:
2108:Introduction to Combinatorial Analysis
1943:(the number of factors of two in the
378:
371:
364:
357:
350:
343:
336:
329:
320:
243:squares in the plane, grouped into a
7:
2121:Peart, Paul; Woan, Wen-Jin (2000),
1787:{\displaystyle G'(x)=G(x)+xG(x)\,.}
744:Summation formula and approximation
18:Number of ways to pair up n objects
2390:Journal of Algebraic Combinatorics
1710:
1707:
1701:
1698:
1695:
1692:
1689:
1318:
999:
815:
656:
653:
650:
529:The telephone numbers satisfy the
14:
2083:, Volume 3: Sorting and Searching
1023:{\displaystyle {\tbinom {n}{2k}}}
257:Robinson–Schensted correspondence
2316:Journal of Computational Biology
2024:, until either reaching zero or
1053:elements to be matched, and the
377:
370:
363:
356:
349:
342:
335:
328:
322:
2662:Journal of Combinatorial Theory
2557:Canadian Journal of Mathematics
2501:Canadian Journal of Mathematics
2081:The Art of Computer Programming
1278:exponential generating function
961:In each term of the first sum,
648:
53:of the fourth telephone number.
1887:
1881:
1855:
1849:
1777:
1771:
1759:
1753:
1744:
1738:
1672:
1660:
1638:
1626:
1598:
1586:
1564:
1552:
1527:
1515:
1493:
1487:
1335:
1329:
1296:
1290:
1235:
1225:
1171:
1165:
1102:
1093:
1078:
1063:
933:
918:
850:
835:
767:
761:
642:
630:
624:
612:
606:
594:
585:
579:
548:
542:
212:-vertex graph is given by the
1:
2806:Factorial and binomial topics
2631:10.1016/S0012-365X(01)00250-3
722:choices for that person, and
2367:10.1016/0012-365X(87)90024-0
2264:Journal of Integer Sequences
2130:Journal of Integer Sequences
1280:of the telephone numbers is
220:vertices is the same as the
700:connection patterns of the
685:first published in 1800 by
273:irreducible representations
119:irreducible representations
2832:
2714:Sloane, N. J. A.
2685:10.1016/j.jcta.2009.08.002
22:
2801:Enumerative combinatorics
2762:10.4310/JOC.2015.v6.n4.a5
2413:10.1007/s10801-014-0534-5
1449:above, multiplying it by
510:Pólya enumeration theorem
187:combinatorial enumeration
109:, the number of standard
2740:Journal of Combinatorics
2599:Bousquet-Mélou, Mireille
2444:The Mathematical Gazette
2329:10.1089/cmb.2005.12.1004
2228:10.1007/1-4020-2634-X_25
2210:Symmetries in Science XI
1430:and, in particular, the
1149:Stirling's approximation
231:A standard Young tableau
93:vertices, the number of
2811:Matching (graph theory)
2718:"Sequence A264737"
2110:, Dover, pp. 85–86
563:{\displaystyle T(0)=1,}
520:Mathematical properties
2571:10.4153/CJM-1955-021-8
2554:in symmetric groups",
2515:10.4153/CJM-1951-038-3
1939:. More precisely, the
1910:
1788:
1718:
1408:
1322:
1263:
1134:
1047:
1024:
975:
955:
896:
807:
679:
564:
232:
224:-th telephone number.
54:
2270:(1), Article 08.1.1,
2136:(2), Article 00.2.1,
2012:For any prime number
1911:
1789:
1719:
1409:
1302:
1264:
1135:
1048:
1025:
976:
956:
862:
773:
687:Heinrich August Rothe
680:
565:
269:representation theory
230:
202:chemical graph theory
127:Heinrich August Rothe
33:
2608:Discrete Mathematics
2353:Discrete Mathematics
2164:Mathematics Magazine
1923:For large values of
1843:
1829:matching polynomials
1727:
1471:
1284:
1159:
1060:
1034:
988:
983:binomial coefficient
965:
755:
573:
536:
485:mathematics of chess
73:that count the ways
71:sequence of integers
2286:2008JIntS..11...11B
2142:2000JIntS...3...21P
1945:prime factorization
1825:Hermite polynomials
1460:, and summing over
1272:Generating function
531:recurrence relation
131:recurrence equation
107:Hermite polynomials
2603:Flajolet, Philippe
2597:Banderier, Cyril;
1906:
1784:
1714:
1622:
1548:
1483:
1404:
1259:
1130:
1046:{\displaystyle 2k}
1043:
1020:
1018:
971:
951:
675:
560:
506:eight rooks puzzle
233:
176:people defines an
101:elements that are
67:involution numbers
55:
51:(4) = 10
2796:Integer sequences
2727:, OEIS Foundation
2601:; Denise, Alain;
2026:detecting a cycle
1901:
1706:
1688:
1679:
1613:
1605:
1539:
1534:
1474:
1394:
1357:
1253:
1221:
1190:
1128:
1011:
974:{\displaystyle k}
946:
827:
661:
476:
475:
63:telephone numbers
2823:
2781:
2780:
2755:
2735:
2729:
2728:
2710:
2704:
2703:
2678:
2669:(8): 1082–1094,
2656:
2650:
2649:
2624:
2594:
2583:
2582:
2573:
2553:
2541:
2535:
2534:
2517:
2488:
2475:
2474:
2450:(404): 131–134,
2439:
2433:
2432:
2415:
2405:
2385:
2379:
2378:
2369:
2346:
2340:
2339:
2323:(7): 1004–1013,
2312:
2303:
2297:
2296:
2279:
2255:
2249:
2248:
2221:
2219:quant-ph/0310174
2204:
2195:
2194:
2159:
2153:
2152:
2127:
2118:
2112:
2111:
2100:
2094:
2093:
2076:Knuth, Donald E.
2072:
2036:
2023:
2019:
2015:
2008:
2001:
1990:
1983:
1972:
1968:
1957:
1938:
1930:
1926:
1915:
1913:
1912:
1907:
1902:
1900:
1899:
1890:
1880:
1879:
1874:
1873:
1862:
1838:
1834:
1827:, which are the
1819:
1812:
1793:
1791:
1790:
1785:
1737:
1723:
1721:
1720:
1715:
1713:
1704:
1686:
1680:
1678:
1658:
1657:
1642:
1621:
1606:
1604:
1584:
1583:
1568:
1547:
1535:
1533:
1513:
1512:
1497:
1482:
1466:
1459:
1448:
1437:
1433:
1429:
1413:
1411:
1410:
1405:
1400:
1396:
1395:
1390:
1389:
1380:
1358:
1356:
1348:
1347:
1346:
1324:
1321:
1316:
1268:
1266:
1265:
1260:
1254:
1252:
1251:
1250:
1246:
1223:
1222:
1217:
1211:
1209:
1208:
1204:
1195:
1191:
1183:
1146:
1139:
1137:
1136:
1131:
1129:
1127:
1119:
1118:
1108:
1091:
1055:double factorial
1052:
1050:
1049:
1044:
1029:
1027:
1026:
1021:
1019:
1017:
1016:
1010:
998:
980:
978:
977:
972:
960:
958:
957:
952:
947:
945:
917:
916:
906:
898:
895:
888:
876:
834:
833:
832:
826:
814:
806:
799:
787:
739:
732:
721:
714:
703:
699:
684:
682:
681:
676:
662:
659:
569:
567:
566:
561:
515:
500:
490:
381:
380:
374:
373:
367:
366:
360:
359:
353:
352:
346:
345:
339:
338:
332:
331:
326:
325:
285:
266:
254:
242:
223:
219:
211:
175:
167:
148:
139:
116:
100:
92:
76:
52:
45:
25:Telephone number
2831:
2830:
2826:
2825:
2824:
2822:
2821:
2820:
2786:
2785:
2784:
2737:
2736:
2732:
2712:
2711:
2707:
2658:
2657:
2653:
2596:
2595:
2586:
2548:
2543:
2542:
2538:
2496:Herstein, I. N.
2490:
2489:
2478:
2456:10.2307/3617799
2441:
2440:
2436:
2387:
2386:
2382:
2349:
2347:
2343:
2310:
2305:
2304:
2300:
2257:
2256:
2252:
2238:
2206:
2205:
2198:
2176:10.2307/2690455
2161:
2160:
2156:
2125:
2120:
2119:
2115:
2102:
2101:
2097:
2074:
2073:
2058:
2054:
2042:
2032:
2021:
2017:
2013:
2003:
1992:
1985:
1974:
1970:
1959:
1948:
1936:
1928:
1924:
1921:
1891:
1864:
1863:
1841:
1840:
1836:
1832:
1814:
1795:
1730:
1725:
1724:
1659:
1643:
1585:
1569:
1514:
1498:
1469:
1468:
1461:
1450:
1439:
1435:
1431:
1419:
1381:
1372:
1368:
1349:
1338:
1325:
1282:
1281:
1274:
1234:
1224:
1212:
1178:
1177:
1157:
1156:
1141:
1110:
1109:
1092:
1058:
1057:
1032:
1031:
1003:
993:
986:
985:
963:
962:
908:
907:
899:
819:
809:
753:
752:
746:
734:
723:
716:
705:
701:
690:
571:
570:
534:
533:
527:
522:
513:
492:
488:
481:
480:
479:
383:
382:
375:
368:
361:
354:
347:
340:
333:
323:
277:symmetric group
264:
252:
240:
237:Ferrers diagram
221:
217:
214:complete graphs
209:
173:
165:
159:
154:
144:
134:
123:symmetric group
114:
98:
90:
74:
47:
44:
38:
28:
19:
12:
11:
5:
2829:
2827:
2819:
2818:
2813:
2808:
2803:
2798:
2788:
2787:
2783:
2782:
2746:(4): 483–508,
2730:
2705:
2651:
2615:(1–3): 29–55,
2584:
2536:
2476:
2434:
2396:(2): 229–255,
2380:
2360:(2): 149–163,
2341:
2298:
2250:
2236:
2196:
2154:
2113:
2095:
2055:
2053:
2050:
2030:
1920:
1917:
1905:
1898:
1894:
1889:
1886:
1883:
1878:
1872:
1869:
1860:
1857:
1854:
1851:
1848:
1783:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1736:
1733:
1712:
1709:
1703:
1700:
1697:
1694:
1691:
1685:
1677:
1674:
1671:
1668:
1665:
1662:
1656:
1653:
1650:
1646:
1640:
1637:
1634:
1631:
1628:
1625:
1620:
1616:
1612:
1609:
1603:
1600:
1597:
1594:
1591:
1588:
1582:
1579:
1576:
1572:
1566:
1563:
1560:
1557:
1554:
1551:
1546:
1542:
1538:
1532:
1529:
1526:
1523:
1520:
1517:
1511:
1508:
1505:
1501:
1495:
1492:
1489:
1486:
1481:
1477:
1403:
1399:
1393:
1388:
1384:
1378:
1375:
1371:
1367:
1364:
1361:
1355:
1352:
1345:
1341:
1337:
1334:
1331:
1328:
1320:
1315:
1312:
1309:
1305:
1301:
1298:
1295:
1292:
1289:
1273:
1270:
1258:
1249:
1245:
1241:
1237:
1233:
1230:
1227:
1220:
1215:
1207:
1203:
1199:
1194:
1189:
1186:
1181:
1176:
1173:
1170:
1167:
1164:
1153:asymptotically
1126:
1123:
1117:
1113:
1107:
1104:
1101:
1098:
1095:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1042:
1039:
1015:
1009:
1006:
1002:
997:
970:
950:
944:
941:
938:
935:
932:
929:
926:
923:
920:
915:
911:
905:
902:
894:
891:
887:
883:
880:
875:
872:
869:
865:
861:
858:
855:
852:
849:
846:
843:
840:
837:
831:
825:
822:
818:
813:
805:
802:
798:
794:
791:
786:
783:
780:
776:
772:
769:
766:
763:
760:
745:
742:
674:
671:
668:
665:
658:
655:
652:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
559:
556:
553:
550:
547:
544:
541:
526:
523:
521:
518:
477:
474:
473:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
443:
440:
436:
435:
432:
428:
427:
424:
420:
419:
416:
412:
411:
408:
404:
403:
400:
396:
395:
392:
388:
387:
384:
376:
369:
362:
355:
348:
341:
334:
327:
321:
319:
315:
314:
312:
309:
306:
303:
300:
297:
294:
291:
288:
283:
282:
261:Young tableaux
158:
155:
142:
111:Young tableaux
87:complete graph
42:
36:complete graph
17:
13:
10:
9:
6:
4:
3:
2:
2828:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2793:
2791:
2779:
2775:
2771:
2767:
2763:
2759:
2754:
2749:
2745:
2741:
2734:
2731:
2726:
2725:
2719:
2715:
2709:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2677:
2672:
2668:
2664:
2663:
2655:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2623:
2618:
2614:
2610:
2609:
2604:
2600:
2593:
2591:
2589:
2585:
2581:
2577:
2572:
2567:
2563:
2559:
2558:
2551:
2546:
2540:
2537:
2533:
2529:
2525:
2521:
2516:
2511:
2507:
2503:
2502:
2497:
2493:
2487:
2485:
2483:
2481:
2477:
2473:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2438:
2435:
2431:
2427:
2423:
2419:
2414:
2409:
2404:
2399:
2395:
2391:
2384:
2381:
2377:
2373:
2368:
2363:
2359:
2355:
2354:
2345:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2317:
2309:
2302:
2299:
2295:
2291:
2287:
2283:
2278:
2273:
2269:
2265:
2261:
2254:
2251:
2247:
2243:
2239:
2237:1-4020-2633-1
2233:
2229:
2225:
2220:
2215:
2211:
2203:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2158:
2155:
2151:
2147:
2143:
2139:
2135:
2131:
2124:
2117:
2114:
2109:
2105:
2104:Riordan, John
2099:
2096:
2092:
2088:
2084:
2082:
2077:
2071:
2069:
2067:
2065:
2063:
2061:
2057:
2051:
2049:
2047:
2040:
2035:
2029:
2027:
2010:
2006:
1999:
1995:
1988:
1981:
1977:
1966:
1962:
1955:
1951:
1946:
1942:
1934:
1919:Prime factors
1918:
1916:
1903:
1896:
1892:
1884:
1876:
1858:
1852:
1846:
1830:
1826:
1821:
1817:
1810:
1806:
1802:
1798:
1781:
1774:
1768:
1765:
1762:
1756:
1750:
1747:
1741:
1734:
1731:
1683:
1675:
1669:
1666:
1663:
1654:
1651:
1648:
1644:
1635:
1632:
1629:
1623:
1618:
1614:
1610:
1607:
1601:
1595:
1592:
1589:
1580:
1577:
1574:
1570:
1561:
1558:
1555:
1549:
1544:
1540:
1536:
1530:
1524:
1521:
1518:
1509:
1506:
1503:
1499:
1490:
1484:
1479:
1475:
1464:
1457:
1453:
1446:
1442:
1427:
1423:
1417:
1416:Taylor series
1401:
1397:
1391:
1386:
1382:
1376:
1373:
1369:
1365:
1362:
1359:
1353:
1350:
1343:
1339:
1332:
1326:
1313:
1310:
1307:
1303:
1299:
1293:
1287:
1279:
1271:
1269:
1256:
1247:
1243:
1239:
1231:
1228:
1218:
1213:
1205:
1201:
1197:
1192:
1187:
1184:
1179:
1174:
1168:
1162:
1154:
1150:
1145:
1124:
1121:
1115:
1111:
1105:
1099:
1096:
1087:
1084:
1081:
1075:
1072:
1069:
1066:
1056:
1040:
1037:
1007:
1004:
1000:
984:
968:
948:
942:
939:
936:
930:
927:
924:
921:
913:
909:
903:
900:
889:
885:
881:
873:
870:
867:
863:
859:
856:
853:
847:
844:
841:
838:
823:
820:
816:
800:
796:
792:
784:
781:
778:
774:
770:
764:
758:
751:
743:
741:
737:
730:
726:
719:
712:
708:
697:
693:
688:
672:
669:
666:
663:
645:
639:
636:
633:
627:
621:
618:
615:
609:
603:
600:
597:
591:
588:
582:
576:
557:
554:
551:
545:
539:
532:
524:
519:
517:
511:
507:
503:
499:
495:
486:
472:
469:
466:
463:
460:
457:
454:
451:
448:
446:
445:
441:
438:
437:
433:
430:
429:
425:
422:
421:
417:
414:
413:
409:
406:
405:
401:
398:
397:
393:
390:
389:
385:
317:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
287:
286:
281:
278:
274:
270:
262:
258:
250:
249:Young tableau
246:
238:
229:
225:
215:
207:
203:
199:
195:
190:
188:
183:
179:
170:
163:
156:
152:
147:
141:
137:
132:
129:, who gave a
128:
124:
120:
112:
108:
104:
96:
88:
84:
80:
72:
68:
64:
60:
50:
41:
37:
32:
26:
21:
16:
2816:Permutations
2743:
2739:
2733:
2721:
2708:
2666:
2665:, Series A,
2660:
2654:
2622:math/0411250
2612:
2606:
2561:
2555:
2549:
2539:
2505:
2499:
2447:
2443:
2437:
2393:
2389:
2383:
2357:
2351:
2344:
2320:
2314:
2301:
2267:
2263:
2253:
2209:
2170:(1): 45–53,
2167:
2163:
2157:
2133:
2129:
2116:
2107:
2098:
2079:
2045:
2043:
2011:
2004:
1997:
1993:
1986:
1979:
1975:
1964:
1960:
1953:
1949:
1941:2-adic order
1933:power of two
1922:
1822:
1815:
1808:
1804:
1800:
1796:
1462:
1455:
1451:
1444:
1440:
1425:
1421:
1275:
1143:
747:
735:
728:
724:
717:
710:
706:
695:
691:
528:
497:
493:
491:rooks on an
482:
267:squares. In
234:
206:Hosoya index
194:graph theory
191:
171:
162:John Riordan
160:
157:Applications
135:
95:permutations
83:Hosoya index
66:
62:
56:
48:
39:
20:
15:
2564:: 159–168,
2508:: 328–334,
2046:inefficient
182:permutation
103:involutions
59:mathematics
2790:Categories
2545:Moser, Leo
2492:Chowla, S.
2052:References
1991:, and for
525:Recurrence
502:chessboard
178:involution
2778:119708272
2753:1406.2356
2676:0902.4311
2532:123802787
2472:250441965
2403:1302.6150
2277:0802.0075
1667:−
1652:−
1633:−
1615:∑
1593:−
1578:−
1559:−
1541:∑
1522:−
1507:−
1476:∑
1366:
1319:∞
1304:∑
1175:∼
1073:−
925:−
893:⌋
879:⌊
864:∑
845:−
804:⌋
790:⌊
775:∑
750:summation
667:≥
637:−
619:−
601:−
245:polyomino
79:matchings
2701:17457503
2647:14804110
2337:16201918
2106:(2002),
2078:(1973),
1803:) ∝ exp(
1735:′
198:matching
2770:3382606
2716:(ed.),
2693:2677675
2639:1884885
2580:0068564
2524:0041849
2464:3617799
2430:7419411
2422:3306071
2376:0913181
2294:2377567
2282:Bibcode
2246:5702844
2192:1092195
2184:2690455
2150:1778992
2138:Bibcode
2091:0445948
2037:in the
2034:A264737
1958:and of
1818:(0) = 1
483:In the
275:of the
149:in the
146:A000085
121:of the
85:) of a
69:form a
65:or the
2776:
2768:
2699:
2691:
2645:
2637:
2578:
2530:
2522:
2470:
2462:
2428:
2420:
2374:
2335:
2292:
2244:
2234:
2190:
2182:
2148:
2089:
2002:it is
1984:it is
1973:; for
1927:, the
1813:, and
1705:
1687:
1467:gives
1151:that,
660:
61:, the
2774:S2CID
2748:arXiv
2697:S2CID
2671:arXiv
2643:S2CID
2617:arXiv
2528:S2CID
2468:S2CID
2460:JSTOR
2426:S2CID
2398:arXiv
2311:(PDF)
2272:arXiv
2242:S2CID
2214:arXiv
2180:JSTOR
2126:(PDF)
1947:) of
1458:− 1)!
113:with
81:(the
2722:The
2333:PMID
2232:ISBN
2039:OEIS
2000:+ 3)
1982:+ 2)
1967:+ 1)
1420:exp(
1276:The
731:− 2)
713:− 1)
180:, a
151:OEIS
34:The
2758:doi
2681:doi
2667:117
2627:doi
2613:246
2566:doi
2552:= 1
2510:doi
2452:doi
2408:doi
2362:doi
2325:doi
2224:doi
2172:doi
2007:+ 2
1989:+ 1
1969:is
1811:/2)
1465:≥ 1
1454:/ (
1428:/2)
1418:of
1363:exp
738:− 2
720:− 1
192:In
138:= 0
97:on
89:on
57:In
2792::
2772:,
2766:MR
2764:,
2756:,
2742:,
2720:,
2695:,
2689:MR
2687:,
2679:,
2641:,
2635:MR
2633:,
2625:,
2611:,
2587:^
2576:MR
2574:,
2560:,
2526:,
2520:MR
2518:,
2504:,
2494:;
2479:^
2466:,
2458:,
2448:58
2446:,
2424:,
2418:MR
2416:,
2406:,
2394:41
2392:,
2372:MR
2370:,
2358:67
2356:,
2331:,
2321:12
2319:,
2313:,
2290:MR
2288:,
2280:,
2268:11
2266:,
2262:,
2240:,
2230:,
2222:,
2199:^
2188:MR
2186:,
2178:,
2168:64
2166:,
2146:MR
2144:,
2132:,
2128:,
2087:MR
2059:^
2009:.
1996:(4
1978:(4
1963:(4
1952:(4
1935:,
1807:+
1424:+
1155:,
496:×
235:A
153:).
140:)
2760::
2750::
2744:6
2683::
2673::
2629::
2619::
2568::
2562:7
2550:x
2512::
2506:3
2454::
2410::
2400::
2364::
2327::
2284::
2274::
2226::
2216::
2174::
2140::
2134:3
2041:)
2022:p
2018:p
2014:p
2005:k
1998:k
1994:T
1987:k
1980:k
1976:T
1971:k
1965:k
1961:T
1956:)
1954:k
1950:T
1937:2
1929:n
1925:n
1904:.
1897:n
1893:i
1888:)
1885:i
1882:(
1877:n
1871:e
1868:H
1859:=
1856:)
1853:n
1850:(
1847:T
1837:n
1833:n
1816:T
1809:x
1805:x
1801:x
1799:(
1797:G
1782:.
1778:)
1775:x
1772:(
1769:G
1766:x
1763:+
1760:)
1757:x
1754:(
1751:G
1748:=
1745:)
1742:x
1739:(
1732:G
1711:s
1708:i
1702:h
1699:c
1696:i
1693:h
1690:w
1684:,
1676:!
1673:)
1670:2
1664:n
1661:(
1655:2
1649:n
1645:x
1639:)
1636:2
1630:n
1627:(
1624:T
1619:n
1611:x
1608:+
1602:!
1599:)
1596:1
1590:n
1587:(
1581:1
1575:n
1571:x
1565:)
1562:1
1556:n
1553:(
1550:T
1545:n
1537:=
1531:!
1528:)
1525:1
1519:n
1516:(
1510:1
1504:n
1500:x
1494:)
1491:n
1488:(
1485:T
1480:n
1463:n
1456:n
1452:x
1447:)
1445:n
1443:(
1441:T
1436:n
1432:n
1426:x
1422:x
1402:.
1398:)
1392:2
1387:2
1383:x
1377:+
1374:x
1370:(
1360:=
1354:!
1351:n
1344:n
1340:x
1336:)
1333:n
1330:(
1327:T
1314:0
1311:=
1308:n
1300:=
1297:)
1294:x
1291:(
1288:G
1257:.
1248:4
1244:/
1240:1
1236:)
1232:e
1229:4
1226:(
1219:n
1214:e
1206:2
1202:/
1198:n
1193:)
1188:e
1185:n
1180:(
1172:)
1169:n
1166:(
1163:T
1144:k
1142:2
1125:!
1122:k
1116:k
1112:2
1106:!
1103:)
1100:k
1097:2
1094:(
1088:=
1085:!
1082:!
1079:)
1076:1
1070:k
1067:2
1064:(
1041:k
1038:2
1014:)
1008:k
1005:2
1001:n
996:(
969:k
949:.
943:!
940:k
937:!
934:)
931:k
928:2
922:n
919:(
914:k
910:2
904:!
901:n
890:2
886:/
882:n
874:0
871:=
868:k
860:=
857:!
854:!
851:)
848:1
842:k
839:2
836:(
830:)
824:k
821:2
817:n
812:(
801:2
797:/
793:n
785:0
782:=
779:k
771:=
768:)
765:n
762:(
759:T
736:n
729:n
727:(
725:T
718:n
711:n
709:(
707:T
702:n
698:)
696:n
694:(
692:T
673:,
670:1
664:n
657:r
654:o
651:f
646:,
643:)
640:2
634:n
631:(
628:T
625:)
622:1
616:n
613:(
610:+
607:)
604:1
598:n
595:(
592:T
589:=
586:)
583:n
580:(
577:T
558:,
555:1
552:=
549:)
546:0
543:(
540:T
514:n
498:n
494:n
489:n
470:h
467:g
464:f
461:e
458:d
455:c
452:b
449:a
442:1
439:1
434:2
431:2
426:3
423:3
418:4
415:4
410:5
407:5
402:6
399:6
394:7
391:7
386:8
318:8
311:h
308:g
305:f
302:e
299:d
296:c
293:b
290:a
265:n
253:n
241:n
222:n
218:n
210:n
174:n
166:n
136:n
115:n
99:n
91:n
75:n
49:T
43:4
40:K
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
