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

1, 12, 21, 123, 132, 213, 231, 312, 321, 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321, 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425

Offset: 1

Author

N. J. A. Sloane and Clark Kimberling

Keywords

• nonn
• easy
• base

Comments

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

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

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]

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

References

• 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).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..5913
• OEIS Wiki, Discussion about alternate definition(s) of this sequence, started by _M. F. Hasler_, Jan 28 2013
• Index entries for sequences related to permutations
• Index entries for sequences which agree for a long time but are different

Crossrefs

A007489(n) gives the position (index) of the term corresponding to last permutation of n elements: (n,n-1,...,1).

The first differences A220664 has interesting fractal structure, see A219664 and A217626.

Cf. also A030298, A055089, A060117, A181073, A352991 (by concatenation).

See A240763 for preferential arrangements.

Programs

Maple

seq(seq(add(s[i]*10^(m-i),i=1..m),s=combinat:-permute([$1..m])),m=1..5); # Robert Israel, Oct 14 2015

Mathematica

Flatten @ Table[FromDigits /@ Permutations[Table[i,{i,n}]],{n,9}] (* For first 409113 terms; Zak Seidov, Oct 03 2015 *)

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
(MIT/GNU Scheme)
;; Antti Karttunen, Dec 18 2012
;; Requires also code from A030298 and A055089:
(define (A030299 n) (vector->base-k (A030298permvec (A084556 n) (A220660 n)) 10))
(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)))))))

Python

from itertools import permutations
def pmap(s, m): return sum(s[i-1]*10**(m-i) for i in range(1, len(s)+1))
def agen():
  m = 1
  while True:
    for s in permutations(range(1, m+1)): yield pmap(s, m)
    m += 1
def aupton(terms):
  alst, g = [], agen()
  while len(alst) < terms: alst += [next(g)]
  return alst
print(aupton(42)) # Michael S. Branicky, Jan 12 2021

Extensions

Edited by N. J. A. Sloane, Feb 23 2010