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This sequence has had and could have several alternative "definitions" or NAMES:

  1. Permutations of lengths 1, 2, 3, ... arranged lexicographically.
  2. Numbers whose decimal representation is a permutation of (1,2,3,4,...,i) for some i >= 1.
  3. Decimal representation of permutations of lengths 1, 2, 3, ... arranged lexicographically.
  4. Numbers N(s) = sum_{i=1..m} s(i)*10^(m-i), where s runs over the permutations of (1,...,m), and m=1,2,3,....

(You may add other definitions above, without making any change to those already given.)

All agree that the first 409113 = A007489(9) terms are

1, 12, 21, 123, 132, ..., 321, 1234, ..., 987654321,

but there is a big ? for the term following 987654321 = A030299(409113).

My suggestion

I favour the 4th definition above. It can actually be seen as an interpretation of the 3rd NAME listed above, if "decimal representation" is defined accordingly (as this function N) for the case of n > 9 in which it has no "predefined" meaning.

I am aware that this is in contradiction with the comment

This is very clumsy once the length exceeds 9. For example, after 987654321 (= A030299(409113), where 409113 = A007489(9)) we get 12345678910, 12345678109,... (which clarifies the precise meaning of "arranged lexicographically", namely: before concatenation). In A030298 this problem has been avoided, by listing the elements of permutations as separate terms.

from the original author NJAS, which I have myself amended, made precise and completed with references some time ago, but would now rather suggest to rephrase as "A possible variant would be...."

The advantage of this 4th definition is that

  1. it is completely well defined for any arbitrarily large index,
  2. it has (IMO) the best possible "mathematical" properties for use in formulas doing arithmetics (differences, divisions...) with the terms
  3. it preserves the fractional pattern of the first differences A220664 divided by 9, with the same subsequences repeating at the same periodic intervals, indefinitely (cf. A219664 and A217626).

Open questions

Open questions are:

  1. Is it useful to have several sequences with so many initial terms being equal ?
    (IMO: not sure, maybe comments about alternate versions would suffice)
  2. Are we sure that it has always be used as it was initially defined ?
    (here I'm actually certain the answer is no !)
  3. Which definition would be the most useful one?
  4. If we decide to have the different versions all individually in OEIS, is it best to keep the existing A-number for the sequence as initially defined (if ever it has been referenced in the precise sense of that definition), or for that which has effectively been used most often? (or maybe even: always?)

In other cases I would be in favour of keeping the original definition and creating new sequences for potentially more "useful" variants. But I am convinced that here, since the sequence has now frequently been used in the sense of definition 4, including in mathematical formulae and statements about divisibility of fist differences etc, and probably in no formula is was used in the sense of A030299(409114) = 12345678910, I'd be in favour of making precise that possible (precise and well defined) interpretation of the (actually: current) definition (which is 3).

Minor issue: lexicographically ordered

I think this would need to be more precise (bewond what I did in NJAS's comment) to be well defined. Since lexicographically, not only would come 10123456789 before 12435678910, but also "123" would come before "21", etc. I think the proposed definition 4 makes clear the "primary" criterium for ordering is m=1,2,3,..., and the secondary is the (implicit) lex order of the permutations s={(1,2,...,m), ..., (m,...,2,1)}, which is at the same time also the strictly increasing order of the terms, since in view of the purely arithmetical definition, a "higher" permutation also results in a greater number.

M. F. Hasler 13:31, 28 January 2013 (UTC)