I came across this chart in an 200 year old publication..
and was wasting my time figuring out by hand the laws by which it is
governed just because i couldn't use J, which seems a perfect fit for
it!
111,112,113,114,115;122,123,124,125;133,134,135;144,145;155
121,131,141,151;212,132,142,152;313,143,153;414,154;515
211,311,411,511;221,213,214,215;331,314,315;441,415;551
231,241,251; 341,351; 451;
312,412,512; 413,513; 514;
321,421,521; 431,531; 541;
222,223,224,225;233,234,235;244,245;255
232,242,252;323,243,253;424,254;525
322,422,522;332,324,325;442,425;552
342,352; 452;
423,523; 524;
432,532; 542;
333,334,335;344,345;355
343,353;434,354;535
433,533;443,435;553
453;
534;
543;
444,445;455
454;545
544;554
555;
it's just the combinations with repetitions of 3 out of 5 elements in
lexicographic order, with the permutations written below the cases.
1 3 3 3 3 3 3|3 6 6 6 6 6 3 6 6 6 6 3 6 6 6 3 6 6 3 6 3 ... 7 [28]
(3 out of 7 elements)
7
1 3 3 3 3 3|3 6 6 6 6 3 6 6 6 3 6 6 3 6 3 ... 6 [21]
(3 out of 6 elements)
6
1 3 3 3 3|3 6 6 6 3 6 6 3 6 3 Multinomial coefficient 5 [15] (3
out of 5 elements)
5
1 3 3 3|3 6 6 3 6 3 follows Pascal Simplex [10] (3 out
of 4 elements)
4
1 3 3|3 6 3 follows Pascal's Tetrahedron(last row excl.)[6]
(3 out of 3 elements)
3
1 3|3 follows Pascal's Triangle [3] (3 out of
2 elements)
2
1 [1] (3
out of 1 element)
number of combinations beginning with the 1st element - triangular numbers
picking 4 elements with repetition would be governed by tetrahedral numbers
and so on
HOW COULD ONE GENERATE ALL THIS IN J??
thanks in advance
atto
Lexicographic Multiset Permutation Generation including Ranking etc.
A New Method for Generating Permutations in Lexicographic Order
http://203.72.2.115/Ejournal/AL03050402.pdf
>From Permutations to Iterative Permutations
http://www.ijcset.net/docs/Volumes/volume2issue7/ijcset2012020702.pdf
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