A030299 -id:A030299 - cp's OEIS frontend

A213059 Subsets of positive integers arranged in canonical order.

1, 12, 2, 123, 13, 23, 3, 1234, 124, 134, 234, 14, 24, 34, 4, 12345, 1235, 1245, 1345, 2345, 125, 135, 145, 235, 245, 345, 15, 25, 35, 45, 5, 123456, 12346, 12356, 12456, 13456, 23456, 1236, 1246, 1256, 1346, 1356, 1456, 2346, 2356, 2456, 3456, 126, 136, 146, 156, 236, 246, 256, 346, 356, 456, 16, 26, 36, 46, 56, 6

Offset: 1

N. J. A. Sloane, Jun 03 2012

nonn

The order is self-explanatory (or see the Kubo-Vakil paper).
Of course once we reach subsets containing 10 this way of representing subsets by concatenation is unsatisfactory. Still, the sequence serves as a pointer to the Kubo-Vakil paper.
Sort by largest element, then decreasing size, then lexicographically (see Kubo-Vakil paper). - Michael S. Branicky, Jan 12 2021

Michael S. Branicky, Table of n, a(n) for n = 1..10000
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252. a(n) is the concatenation of their S(n).

Cf. A030299.

from itertools import chain, combinations as C
def powerset(s): # in decreasing size
  return chain.from_iterable(C(s, r) for r in range(len(s), -1, -1))
def agen():
  m = 1 # largest element
  while True:
    for p in powerset(range(1, m)): yield int("".join(map(str, p+(m,))))
    m += 1
def aupton(terms):
  alst, g = [], agen()
  while len(alst) < terms: alst += [next(g)]
  return alst
print(aupton(63)) # Michael S. Branicky, Jan 12 2021

a(25) corrected by Michael S. Branicky, Jan 12 2021

A268532 Decimal representation of permutations of lengths 1, 2, 3, ... arranged first by number of inversions and then lexicographically.

Original entry on oeis.org

1, 12, 21, 123, 132, 213, 231, 312, 321, 1234, 1243, 1324, 2134, 1342, 1423, 2143, 2314, 3124, 1432, 2341, 2413, 3142, 3214, 4123, 2431, 3241, 3412, 4132, 4213, 3421, 4231, 4312, 4321, 12345, 12354, 12435, 13245, 21345, 12453

Offset: 1

N. J. A. Sloane, Feb 19 2016

			The 24 permutations of length 4 and their numbers of inversions:
1234 0
1243 1
1324 1
2134 1
1342 2
1423 2
2143 2
2314 2
3124 2
1432 3
2341 3
2413 3
3142 3
3214 3
4123 3
2431 4
3241 4
3412 4
4132 4
4213 4
3421 5
4231 5
4312 5
4321 6

David Consiglio, Jr., Table of n, a(n) for n = 1..5913
David Consiglio, Jr., Python Program

Cf. A000142 (n!), A030299 (in purely lexicographic order), A008302 (number of permutations with k inversions).

A320588 Derangements of {1,2,...,n} (n >= 2) in lexicographic order.

Original entry on oeis.org

21, 231, 312, 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321, 21453, 21534, 23154, 23451, 23514, 24153, 24513, 24531, 25134, 25413, 25431, 31254, 31452, 31524, 34152, 34251, 34512, 34521, 35124, 35214, 35412, 35421, 41253, 41523, 41532, 43152, 43251, 43512

Offset: 2

Enrique Navarrete, Nov 14 2018

The number of derangements of {1,2,...,n} is given in A000166.

This sequence is unsatisfactory for n >= 10. To have a sequence that is defined for all n, the derangements should be comma-separated lists, with keyword tabf. - N. J. A. Sloane, Nov 15 2018

			Triangle begins:
     21;
    231,   312;
   2143,  2341,  2413,  3142,  3412,  3421,  4123,  4312,  4321;
  21453, 21534, 23154, 23451, 23514, 24153, 24513, 24531, 25134, ...
  ...
From _David A. Corneth_, Nov 15 2018: (Start)
43512 is in the sequence because no digit is equal to the index of the digit in the number (with offset 1).
43125 is not in the sequence because 5 is at the fifth position. (End)

Cf. A000166, A030299.

Cf. A214261.

Needs["Combinatorica`"]; Flatten @ Table[FromDigits /@ Derangements [Table[i, {i, n}]], {n, 9}] (* For first 150504 terms, Amiram Eldar, Nov 15 2018 after Zak Seidov at A030299 *)

use ntheory ":all"; my(@L,@d); do { @d=(1..$); forderange { push @L,join"",@d[@]; } $; } for 2..6; say join ",",@L; # _Dana Jacobsen, Nov 15 2018

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A213059 Subsets of positive integers arranged in canonical order.

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A268532 Decimal representation of permutations of lengths 1, 2, 3, ... arranged first by number of inversions and then lexicographically.

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A320588 Derangements of {1,2,...,n} (n >= 2) in lexicographic order.

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