A175061 -id:A175061 - cp's OEIS frontend

A347759 Let s(k) be the finite permutation associated to A175061(k); s(a(n)) is the inverse of s(n).

1, 2, 3, 4, 5, 9, 7, 8, 6, 13, 11, 12, 10, 14, 15, 27, 33, 26, 32, 20, 21, 22, 28, 24, 30, 18, 16, 23, 29, 25, 31, 19, 17, 50, 48, 36, 42, 38, 44, 51, 49, 37, 43, 39, 45, 46, 47, 35, 41, 34, 40, 52, 53, 57, 55, 56, 54, 104, 102, 150, 126, 152, 128, 105, 103

Offset: 1

Rémy Sigrist, Sep 12 2021

This sequence is a self-inverse permutation of the positive integers.

			For n = 42:
- the binary expansion of A175061(42) is "100001110000011",
- so s(42) = [1, 4, 3, 5, 2],
- and s(a(42)) = [1, 5, 3, 2, 4],
- and the binary expansion of A175061(a(42)) = "100000111001111",
- and a(42) = 37.

Rémy Sigrist, Table of n, a(n) for n = 1..5913
Rémy Sigrist, PARI program for A347759
Index entries for sequences that are permutations of the natural numbers

Cf. A003422, A175061, A347758.

PARI
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See Links section.
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a(n) < A003422(k) for any n < A003422(k).

A175062 An arrangement of permutations. Irregular table read by rows: Read A175061(n) in binary from left to right. Row n contains the lengths of the runs of 0's and 1's.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 3, 2, 1, 2, 3, 2, 3, 1, 2, 1, 3, 3, 2, 1, 3, 1, 2, 1, 4, 2, 3, 1, 4, 3, 2, 1, 3, 2, 4, 1, 3, 4, 2, 1, 2, 3, 4, 1, 2, 4, 3, 2, 4, 1, 3, 2, 4, 3, 1, 2, 3, 1, 4, 2, 3, 4, 1, 2, 1, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 3, 4, 2, 1, 3, 2, 1, 4, 3, 2, 4, 1, 3, 1, 2, 4, 3, 1, 4, 2, 4, 3, 1, 2, 4, 3, 2, 1, 4, 2

Offset: 1

Leroy Quet, Dec 12 2009

Let F(n) = sum{k=1 to n} k!. Then rows F(n-1)+1 to F(n) are the permutations of (1,2,3,...,n). (And each row in this range is made up of exactly n terms, obviously.)

			A175061(10) = 536 in binary is 1000011000. This contains a run of one 1, followed by a run of four 0's, followed by a run of two 1's, followed finally by a run of three 0's. So row 10 consists of the run lengths (1,4,2,3), a permutation of (1,2,3,4).

Cf. A175061

Extended by Ray Chandler, Dec 16 2009

A109299 Primal codes of canonical finite permutations on positive integers.

Original entry on oeis.org

1, 2, 12, 18, 360, 540, 600, 1350, 1500, 2250, 75600, 105840, 113400, 126000, 158760, 246960, 283500, 294000, 315000, 411600, 472500, 555660, 735000, 864360, 992250, 1296540, 1389150, 1440600, 1653750, 2572500, 3241350, 3601500, 3858750

Offset: 1

Jon Awbrey, Jul 09 2005

nonn

A canonical finite permutation on positive integers is a bijective mapping of [n] = {1, ..., n} to itself, counting the empty mapping as a permutation of the empty set.
From Rémy Sigrist, Sep 18 2021: (Start)
As usual with lists, the terms of the sequence are given in ascending order.
Equivalently, these are the numbers m such that A001221(m) = A051903(m) = A061395(m) = A071625(m).
This sequence has connections with A175061; here the prime factorizations, there the run-lengths in binary expansions, encode finite permutations.
There are m! terms with m distinct prime factors, the least one being A006939(m) and the greatest one being A076954(m); these m! terms are not necessarily contiguous. (End)

			Writing (prime(i))^j as i:j, we have this table:
Primal Codes of Canonical Finite Permutations
        1 = { }
        2 = 1:1
       12 = 1:2 2:1
       18 = 1:1 2:2
      360 = 1:3 2:2 3:1
      540 = 1:2 2:3 3:1
      600 = 1:3 2:1 3:2
     1350 = 1:1 2:3 3:2
     1500 = 1:2 2:1 3:3
     2250 = 1:1 2:2 3:3
    75600 = 1:4 2:3 3:2 4:1
   105840 = 1:4 2:3 3:1 4:2
   113400 = 1:3 2:4 3:2 4:1
   126000 = 1:4 2:2 3:3 4:1
   158760 = 1:3 2:4 3:1 4:2
   246960 = 1:4 2:2 3:1 4:3
   283500 = 1:2 2:4 3:3 4:1
   294000 = 1:4 2:1 3:3 4:2
   315000 = 1:3 2:2 3:4 4:1
   411600 = 1:4 2:1 3:2 4:3
   472500 = 1:2 2:3 3:4 4:1
   555660 = 1:2 2:4 3:1 4:3
   735000 = 1:3 2:1 3:4 4:2
   864360 = 1:3 2:2 3:1 4:4
   992250 = 1:1 2:4 3:3 4:2
  1296540 = 1:2 2:3 3:1 4:4
  1389150 = 1:1 2:4 3:2 4:3
  1440600 = 1:3 2:1 3:2 4:4
  1653750 = 1:1 2:3 3:4 4:2
  2572500 = 1:2 2:1 3:4 4:3
  3241350 = 1:1 2:3 3:2 4:4
  3601500 = 1:2 2:1 3:3 4:4
  3858750 = 1:1 2:2 3:4 4:3
  5402250 = 1:1 2:2 3:3 4:4

Suggested by Franklin T. Adams-Watters

J. Awbrey, Riffs and Rotes
Rémy Sigrist, PARI program for A109299

Cf. A001221, A006939, A051903, A061395, A071625, A076954, A106177, A108352, A108371, A109297, A109298, A109301, A175061, A347758.

PARI
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\\ See Links section.
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is(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); Set(e)==[1..#e] && (#p==0 || p[#p]==prime(#p)) } \\ Rémy Sigrist, Sep 18 2021

Offset changed to 1 and data corrected by Rémy Sigrist, Sep 18 2021

A175356 Those positive integers n where, when written in binary, there are exactly k number of runs (of either 0's or 1's) each of exactly k length, for all k where 1<=k<=m, for some positive integer m.

Original entry on oeis.org

1, 19, 25, 27, 8984, 8988, 9016, 9100, 9112, 9116, 9784, 10008, 10012, 10040, 12568, 12572, 12600, 12680, 12686, 12728, 12740, 12742, 12744, 12750, 12760, 12764, 12856, 13192, 13198, 13240, 13880, 14104, 14108, 14136, 14476, 14488, 14492, 14532, 14534, 14536

Offset: 1

Leroy Quet, Apr 22 2010

A "run" of 0's is not immediately bounded by any 0's, and a "run" of 1's is not immediately bounded by any 1's.
There are exactly (m*(m+1)/2)! / Product_{k=1 to m} k! numbers in the sequence each of m^3/3 + m^2/2 + m/6 binary digits, for all m >= 1, and none of any other number of binary digits.
A005811(a(n)) is triangular, i.e., in A000217. - Michael S. Branicky, Jan 19 2021

			9016 in binary is 10001100111000. There is exactly one run of one binary digit, two runs of two binary digits, and three runs of three binary digits. (Note that it doesn't matter if the runs are of 0's or of 1's.) So, 9016 is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..12664
Rémy Sigrist, PARI program for A175356

Cf. A000330, A022915, A175061, A175357.

Cf. A005811, A000217.

PARI
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\\ See Links section.
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from itertools import groupby
def ok(n):
  runlens = [len(list(g)) for k, g in groupby(bin(n)[2:])]
  return all(runlens.count(k) == k for k in range(1, max(runlens)+1))
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(14536)) # Michael S. Branicky, Jan 19 2021

More terms from Rémy Sigrist, Feb 06 2019

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A347759 Let s(k) be the finite permutation associated to A175061(k); s(a(n)) is the inverse of s(n).

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A175062 An arrangement of permutations. Irregular table read by rows: Read A175061(n) in binary from left to right. Row n contains the lengths of the runs of 0's and 1's.

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A109299 Primal codes of canonical finite permutations on positive integers.

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A175356 Those positive integers n where, when written in binary, there are exactly k number of runs (of either 0's or 1's) each of exactly k length, for all k where 1<=k<=m, for some positive integer m.

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