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

Author

N. J. A. Sloane, Jun 03 2012

Keywords

• nonn

Comments

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

Links

• 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).

Crossrefs

Cf. A030299.

Programs

Python

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

Extensions

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