Elegant iteration over five dice

Question

Is there an elegant way to iterate through possible dice rolls with up to five dice?

I want to replace this hacky Python:

rolls = [
    [str(a) for a in range(1,7)],
    [''.join([str(a), str(b)])
                 for a in range(1, 7)
                 for b in range(1, 7)
                 if a <= b],
    [''.join([str(a), str(b), str(c)])
                 for a in range(1, 7)
                 for b in range(1, 7)
                 for c in range(1, 7)
                 if a <= b <= c],
    [''.join([str(a), str(b), str(c), str(d)])
                 for a in range(1, 7)
                 for b in range(1, 7)
                 for c in range(1, 7)
                 for d in range(1, 7)
                 if a <= b <= c <= d],
    [''.join([str(a), str(b), str(c), str(d), str(e)])
                 for a in range(1, 7)
                 for b in range(1, 7)
                 for c in range(1, 7)
                 for d in range(1, 7)
                 for e in range(1, 7)
                 if a <= b <= c <= d <= e]
]

print(rolls)

Desired result:

[['1', '2', '3', '4', '5', '6'], 
['11', '12', '13', '14', '15', '16', '22', '23', '24', '25', '26', '33', '34', '35', '36', '44', '45', '46', '55', '56', '66'], 
['111', '112', '113', '114', '115', '116', '122', '123', '124', '125', '126', '133', '134', '135', '136', '144', '145', '146', '155', '156', '166', '222', '223', '224', '225', '226', '233', '234', '235', '236', '244', '245', '246', '255', '256', '266', '333', '334', '335', '336', '344', '345', '346', '355', '356', '366', '444', '445', '446', '455', '456', '466', '555', '556', '566', '666'], 
['1111', '1112', '1113', '1114', '1115', '1116', '1122', '1123', '1124', '1125', '1126', '1133', '1134', '1135', '1136', '1144', '1145', '1146', '1155', '1156', '1166', '1222', '1223', '1224', '1225', '1226', '1233', '1234', '1235', '1236', '1244', '1245', '1246', '1255', '1256', '1266', '1333', '1334', '1335', '1336', '1344', '1345', '1346', '1355', '1356', '1366', '1444', '1445', '1446', '1455', '1456', '1466', '1555', '1556', '1566', '1666', '2222', '2223', '2224', '2225', '2226', '2233', '2234', '2235', '2236', '2244', '2245', '2246', '2255', '2256', '2266', '2333', '2334', '2335', '2336', '2344', '2345', '2346', '2355', '2356', '2366', '2444', '2445', '2446', '2455', '2456', '2466', '2555', '2556', '2566', '2666', '3333', '3334', '3335', '3336', '3344', '3345', '3346', '3355', '3356', '3366', '3444', '3445', '3446', '3455', '3456', '3466', '3555', '3556', '3566', '3666', '4444', '4445', '4446', '4455', '4456', '4466', '4555', '4556', '4566', '4666', '5555', '5556', '5566', '5666', '6666'], 
['11111', '11112', '11113', '11114', '11115', '11116', '11122', '11123', '11124', '11125', '11126', '11133', '11134', '11135', '11136', '11144', '11145', '11146', '11155', '11156', '11166', '11222', '11223', '11224', '11225', '11226', '11233', '11234', '11235', '11236', '11244', '11245', '11246', '11255', '11256', '11266', '11333', '11334', '11335', '11336', '11344', '11345', '11346', '11355', '11356', '11366', '11444', '11445', '11446', '11455', '11456', '11466', '11555', '11556', '11566', '11666', '12222', '12223', '12224', '12225', '12226', '12233', '12234', '12235', '12236', '12244', '12245', '12246', '12255', '12256', '12266', '12333', '12334', '12335', '12336', '12344', '12345', '12346', '12355', '12356', '12366', '12444', '12445', '12446', '12455', '12456', '12466', '12555', '12556', '12566', '12666', '13333', '13334', '13335', '13336', '13344', '13345', '13346', '13355', '13356', '13366', '13444', '13445', '13446', '13455', '13456', '13466', '13555', '13556', '13566', '13666', '14444', '14445', '14446', '14455', '14456', '14466', '14555', '14556', '14566', '14666', '15555', '15556', '15566', '15666', '16666', '22222', '22223', '22224', '22225', '22226', '22233', '22234', '22235', '22236', '22244', '22245', '22246', '22255', '22256', '22266', '22333', '22334', '22335', '22336', '22344', '22345', '22346', '22355', '22356', '22366', '22444', '22445', '22446', '22455', '22456', '22466', '22555', '22556', '22566', '22666', '23333', '23334', '23335', '23336', '23344', '23345', '23346', '23355', '23356', '23366', '23444', '23445', '23446', '23455', '23456', '23466', '23555', '23556', '23566', '23666', '24444', '24445', '24446', '24455', '24456', '24466', '24555', '24556', '24566', '24666', '25555', '25556', '25566', '25666', '26666', '33333', '33334', '33335', '33336', '33344', '33345', '33346', '33355', '33356', '33366', '33444', '33445', '33446', '33455', '33456', '33466', '33555', '33556', '33566', '33666', '34444', '34445', '34446', '34455', '34456', '34466', '34555', '34556', '34566', '34666', '35555', '35556', '35566', '35666', '36666', '44444', '44445', '44446', '44455', '44456', '44466', '44555', '44556', '44566', '44666', '45555', '45556', '45566', '45666', '46666', '55555', '55556', '55566', '55666', '56666', '66666']]

Answer 1

You can use itertools' combinations_with_replacement.

For example with 3 4-sided dice (just because the output isn't too large):

>>> from itertools import combinations_with_replacement

>>> dice = 3
>>> faces = 4
>>> list(combinations_with_replacement(range(1, faces+1), dice))
[(1, 1, 1),
 (1, 1, 2),
 (1, 1, 3),
 (1, 1, 4),
 (1, 2, 2),
 (1, 2, 3),
 (1, 2, 4),
 (1, 3, 3),
 (1, 3, 4),
 (1, 4, 4),
 (2, 2, 2),
 (2, 2, 3),
 (2, 2, 4),
 (2, 3, 3),
 (2, 3, 4),
 (2, 4, 4),
 (3, 3, 3),
 (3, 3, 4),
 (3, 4, 4),
 (4, 4, 4)]