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A030299 Decimal representation of permutations of lengths 1, 2, 3, ... arranged lexicographically.

%I A030299 #65 Jun 28 2025 18:22:56
%S A030299 1,12,21,123,132,213,231,312,321,1234,1243,1324,1342,1423,1432,2134,
%T A030299 2143,2314,2341,2413,2431,3124,3142,3214,3241,3412,3421,4123,4132,
%U A030299 4213,4231,4312,4321,12345,12354,12435,12453,12534,12543,13245,13254,13425
%N A030299 Decimal representation of permutations of lengths 1, 2, 3, ... arranged lexicographically.
%C A030299 This is a list of the permutations in "one-line" notation (cf. Dixon and Mortimer, p. 2). The i-th element of the string is the image of i under the permutation. For example 231 is the permutation that sends 1 to 2, 2 to 3, and 3 to 1. - _N. J. A. Sloane_, Apr 12 2014
%C A030299 Precise definition of the term "Decimal representation" (required for indices n>409113): 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,.... This also defines the "lexicographical" order: Obviously 21 comes before 123, etc. The lexicographical order of the permutations, for given m, is the same as the natural order of the numbers N(s). - _M. F. Hasler_, Jan 28 2013
%C A030299 An alternate variant, using concatenation of the permutations, is very clumsy once the length exceeds 9. For example, after 987654321 (= A030299(409113), where 409113 = A007489(9)) we would get 12345678910, 12345678109, ... In A030298 this problem has been avoided by listing the elements of permutations as separate terms. [Edited by _M. F. Hasler_, Jan 28 2013]
%C A030299 Sequence A051845 is a base-independent version of this sequence: Permutations of 1...m are considered as numbers written in base m+1. - _M. F. Hasler_, Jan 28 2013
%D A030299 John D. Dixon and Brian Mortimer, Permutation groups. Graduate Texts in Mathematics, 163. Springer-Verlag, New York, 1996. xii+346 pp. ISBN: 0-387-94599-7 MR1409812 (98m:20003).
%H A030299 Antti Karttunen, <a href="/A030299/b030299.txt">Table of n, a(n) for n = 1..5913</a>
%H A030299 OEIS Wiki, <a href="/wiki/Talk:A030299">Discussion about alternate definition(s) of this sequence</a>, started by _M. F. Hasler_, Jan 28 2013
%H A030299 <a href="/index/Per#perm">Index entries for sequences related to permutations</a>
%H A030299 <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>
%p A030299 seq(seq(add(s[i]*10^(m-i),i=1..m),s=combinat:-permute([$1..m])),m=1..5); # _Robert Israel_, Oct 14 2015
%t A030299 Flatten @ Table[FromDigits /@ Permutations[Table[i,{i,n}]],{n,9}] (* For first 409113 terms; _Zak Seidov_, Oct 03 2015 *)
%o A030299 (PARI) is_A030299(n)={ (n>1234567890 & print("maybe")) || vecsort(digits(n))==vector(#Str(n),i,i) } \\ /* use digits(n)=eval(Vec(Str(n))) in older versions lacking this function */ \\ _M. F. Hasler_, Dec 12 2012
%o A030299 (MIT/GNU Scheme)
%o A030299 ;; _Antti Karttunen_, Dec 18 2012
%o A030299 ;; Requires also code from A030298 and A055089:
%o A030299 (define (A030299 n) (vector->base-k (A030298permvec (A084556 n) (A220660 n)) 10))
%o A030299 (define (vector->base-k vec k) (let loop ((i 0) (s 0)) (cond ((= (vector-length vec) i) s) ((>= (vector-ref vec i) k) (error (format #f "Cannot interpret vector ~a in base ~a!" vec k))) (else (loop (+ i 1) (+ (* k s) (vector-ref vec i)))))))
%o A030299 (Python)
%o A030299 from itertools import permutations
%o A030299 def pmap(s, m): return sum(s[i-1]*10**(m-i) for i in range(1, len(s)+1))
%o A030299 def agen():
%o A030299   m = 1
%o A030299   while True:
%o A030299     for s in permutations(range(1, m+1)): yield pmap(s, m)
%o A030299     m += 1
%o A030299 def aupton(terms):
%o A030299   alst, g = [], agen()
%o A030299   while len(alst) < terms: alst += [next(g)]
%o A030299   return alst
%o A030299 print(aupton(42)) # _Michael S. Branicky_, Jan 12 2021
%Y A030299 A007489(n) gives the position (index) of the term corresponding to last permutation of n elements: (n,n-1,...,1).
%Y A030299 The first differences A220664 has interesting fractal structure, see A219664 and A217626.
%Y A030299 Cf. also A030298, A055089, A060117, A181073, A352991 (by concatenation).
%Y A030299 See A240763 for preferential arrangements.
%K A030299 nonn,easy,base
%O A030299 1,2
%A A030299 _N. J. A. Sloane_ and _Clark Kimberling_
%E A030299 Edited by _N. J. A. Sloane_, Feb 23 2010