I have a Cage Combinations file which I refer to for anything longer than 3 cells, which I've got to learn over time.
For very large cages, it's often easier to think about which numbers are missing.
2-cell cages
3(2) = {12} 4(2) = {13} 5(2) = {14/23} 6(2) = {15/24} 7(2) = {16/25/34}, no 7,8,9 8(2) = {17/26/35}, no 4,8,9 9(2) = {18/27/36/45}, no 9 10(2) = {19/28/37/46}, no 5 11(2) = {29/38/47/56}, no 1 12(2) = {39/48/57}, no 1,2,6 13(2) = {49/58/67}, no 1,2,3 14(2) = {59/68} 15(2) = {69/78} 16(2) = {79} 17(2) = {89}
3-cell cages
6(3) = {123} 7(3) = {124} 8(3) = {125/134} 9(3) = {126/135/234}, no 7,8,9 10(3) = {127/136/145/235}, no 8,9 11(3) = {128/137/146/236/245}, no 9 12(3) = {129/138/147/156/237/246/345} 13(3) = {139/148/157/238/247/256/346} 14(3) = {149/158/167/239/248/257/347/356} 15(3) = {159/168/249/258/267/348/357/456} 16(3) = {169/178/259/268/349/358/367/457} 17(3) = {179/269/278/359/368/458/467} 18(3) = {189/279/369/378/459/468/567} 19(3) = {289/379/469/478/568}, no 1 20(3) = {389/479/569/578}, no 1,2 21(3) = {489/579/678}, no 1,2,3 22(3) = {589/679} 23(3) = {689} 24(3) = {789}
4-cell cages
10(4) = {1234} 11(4) = {1235} 12(4) = {1236/1245}, no 7,8,9 13(4) = {1237/1246/1345}, no 8,9 14(4) = {1238/1247/1256/1346/2345}, no 9 15(4) = {1239/1248/1257/1347/1356/2346} 16(4) = {1249/1258/1267/1348/1357/1456/2347/2356} 17(4) = {1259/1268/1349/1358/1367/1457/2348/2357/2456} 18(4) = {1269/1278/1359/1368/1458/1467/2349/2358/2367/2457/3456} 19(4) = {1279/1369/1378/1459/1468/1567/2359/2368/2458/2467/3457} 20(4) = {1289/1379/1469/1478/1568/2369/2378/2459/2468/2567/3458/3467} 21(4) = {1389/1479/1569/1578/2379/2469/2478/2568/3459/3468/3567} 22(4) = {1489/1579/1678/2389/2479/2569/2578/3469/3478/3568/4567} 23(4) = {1589/1679/2489/2579/2678/3479/3569/3578/4568} 24(4) = {1689/2589/2679/3489/3579/3678/4569/4578} 25(4) = {1789/2689/3589/3679/4579/4678} 26(4) = {2789/3689/4589/4679/5678}, no 1 27(4) = {3789/4689/5679}, no 1,2 28(4) = {4789/5689}, no 1,2,3 29(4) = {5789} 30(4) = {6789}
5-cell cages
15(5) = {12345} 16(5) = {12346} 17(5) = {12347/12356}, no 8,9 18(5) = {12348/12357/12456}, no 9 19(5) = {12349/12358/12367/12457/13456} 20(5) = {12359/12368/12458/12467/13457/23456} 21(5) = {12369/12378/12459/12468/12567/13458/13467/23457} 22(5) = {12379/12469/12478/12568/13459/13468/13567/23458/23467} 23(5) = {12389/12479/12569/12578/13469/13478/13568/14567/23459/23468/23567} 24(5) = {12489/12579/12678/13479/13569/13578/14568/23469/23478/23568/24567} 25(5) = {12589/12679/13489/13579/13678/14569/14578/23479/23569/23578/24568/34567} 26(5) = {12689/13589/13679/14579/14678/23489/23579/23678/24569/24578/34568} 27(5) = {12789/13689/14589/14679/15678/23589/23679/24579/24678/34569/34578} 28(5) = {13789/14689/15679/23689/24589/24679/25678/34579/34678} 29(5) = {14789/15689/23789/24689/25679/34589/34679/35678} 30(5) = {15789/24789/25689/34689/35679/45678} 31(5) = {16789/25789/34789/35689/45679} 32(5) = {26789/35789/45689}, no 1 33(5) = {36789/45789}, no 1,2 34(5) = {46789} 35(5) = {56789}
6-cell cages
21(6) = {123456} 22(6) = {123457} 23(6) = {123458/123467}, no 9 24(6) = {123459/123468/123567} 25(6) = {123469/123478/123568/124567} 26(6) = {123479/123569/123578/124568/134567} 27(6) = {123489/123579/123678/124569/124578/134568/234567} 28(6) = {123589/123679/124579/124678/134569/134578/234568} 29(6) = {123689/124589/124679/125678/134579/134678/234569/234578} 30(6) = {123789/124689/125679/134589/134679/135678/234579/234678} 31(6) = {124789/125689/134689/135679/145678/234589/234679/235678} 32(6) = {125789/134789/135689/145679/234689/235679/245678} 33(6) = {126789/135789/145689/234789/235689/245679/345678} 34(6) = {136789/145789/235789/245689/345679} 35(6) = {146789/236789/245789/345689} 36(6) = {156789/246789/345789} 37(6) = {256789/346789}, no 1 38(6) = {356789} 39(6) = {456789}
7-cell cages
28(7) = {1234567} 29(7) = {1234568} 30(7) = {1234569/1234578} 31(7) = {1234579/1234678} 32(7) = {1234589/1234679/1235678} 33(7) = {1234689/1235679/1245678} 34(7) = {1234789/1235689/1245679/1345678} 35(7) = {1235789/1245689/1345679/2345678} 36(7) = {1236789/1245789/1345689/2345679} 37(7) = {1246789/1345789/2345689} 38(7) = {1256789/1346789/2345789} 39(7) = {1356789/2346789} 40(7) = {1456789/2356789} 41(7) = {2456789} 42(7) = {3456789}
8-cell cages
36(8) = {12345678} 37(8) = {12345679} 38(8) = {12345689} 39(8) = {12345789} 40(8) = {12346789} 41(8) = {12356789} 42(8) = {12456789} 43(8) = {13456789} 44(8) = {23456789}
and finally
45(9) = {123456789}
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