A263329 -id:A263329 - cp's OEIS frontend

A263327 A permutation of {0, 1, ..., 1023} corresponding to lexicographical ordering A262557 of numbers with decreasing digits A009995.

0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 65, 66, 68, 72, 80, 96, 129, 130, 132, 136, 144, 160, 192, 257, 258, 260, 264, 272, 288, 320, 384, 513, 514, 516, 520, 528, 544, 576, 640, 768, 7, 11, 13, 14

Offset: 0

Reinhard Zumkeller, Oct 15 2015

For n = 1..1023, A262557(a(n)) = A009995(n).

Cycle type = (1^12, 3^2, 10^2, 74, 912), i.e., this permutation has 12 fixed points, two 3-cycles and two 10-cycles, and two more cycles of length 74 and 912. See A263355 for the list of these cycles, A263383 for the length of the n-th cycle (ordered by increasing largest element).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1023

Cf. A009995, A262557, A263328 (inverse), A263329 (fixed points), A263383, A263355 (cycles).

Row 10 of A294648.

a263327 0 = 0
a263327 n = head [x | x <- [1..1023], a262557 x == a009995' n]

SortBy[Range[0, 1023], DigitCount[#, 2, 1] &] (* Paolo Xausa, Mar 31 2025 *)

A263327=vecsort(A262557,,1) \\ Does not include a(0)=0. - M. F. Hasler, Dec 11 2019

Edited by M. F. Hasler, Dec 11 2019

A263328 Permutation of {0, ..., 1023} corresponding to lexicographic ordering of numbers with decreasing digits (A009995). Inverse of A263327.

Original entry on oeis.org

0, 1, 2, 11, 3, 12, 13, 56, 4, 14, 15, 57, 16, 58, 59, 176, 5, 17, 18, 60, 19, 61, 62, 177, 20, 63, 64, 178, 65, 179, 180, 386, 6, 21, 22, 66, 23, 67, 68, 181, 24, 69, 70, 182, 71, 183, 184, 387, 25, 72, 73, 185, 74, 186, 187, 388, 75, 188, 189, 389, 190

Offset: 0

Reinhard Zumkeller, Oct 15 2015

For n = 1..1023: A009995(a(n)) = A262557(n).

The fixed points and cycles of this permutation (up to reversal the same as for its inverse A263327) are listed in A263355. - M. F. Hasler, Dec 11 2019

Reinhard Zumkeller, Table of n, a(n) for n = 0..1023

Cf. A009995, A262557, A263327 (inverse), A263329 (fixed points).

a263328 0 = 0
a263328 n = head [x | x <- [1..1023], a009995' x == a262557 n]

A263328=vecsort(A263327,,1) \\ Does not include a(0)=0. - M. F. Hasler, Dec 11 2019

A263355 Table read by rows: cycles of the permutation A263327, sorted in increasing order of their largest element. The elements in each cycle are listed in decreasing numerical order.

Original entry on oeis.org

0, 1, 2, 16, 12, 5, 17, 18, 84, 192, 75, 68, 65, 64, 56, 38, 28, 26, 7, 939, 978, 908, 881, 853, 852, 840, 809, 798, 782, 777, 776, 772, 760, 758, 756, 746, 736, 717, 711, 708, 703, 698, 690, 669, 666, 662, 647, 622, 610, 595, 585, 564, 555, 553, 547, 531

Offset: 1

Reinhard Zumkeller, Oct 16 2015

A263383(n) gives the number of terms in row n.
Fixed points: T(k,m) in A263329 <=> A263383(k) = 1 = m. [Corrected by M. F. Hasler, Dec 11 2019]
The permutations A263327 and its inverse A263328 have 18 cycles, of which 12 are fixed points (listed in A263329), two are 3-cycles (rows 4 and 14 of this table), two are 10-cycles (rows 8 & 13), one is a 74-cycle (row 10) and one is a 912-cycle. - M. F. Hasler, Dec 11 2019
Normally one would list the elements in each cycle in the order in which they appear when the permutation is applied, but that is not the order used here. - N. J. A. Sloane, Dec 11 2019

			   n | Cycles: A263355(n, k=1..A263383(n))                    | A263383(n)
  ---+--------------------------------------------------------+-----------
   1 | (0)                                                    |       1
   2 | (1)                                                    |       1
   3 | (2)                                                    |       1
   4 | (16, 12, 5)                                            |       3
   5 | (17)                                                   |       1
   6 | (18)                                                   |       1
   7 | (84)                                                   |       1
   8 | (192, 75, 68, 65, 64, 56, 38, 28, 26, 7)               |      10
   9 | (939)                                                  |       1
  10 | (978, 908, 881, 853, 852, 840, ..., 142, 115, 45)      |      74
  11 | (1005)                                                 |       1
  12 | (1006)                                                 |       1
  13 | (1016, 997, 995, 985, 967, 959, 958, 955, 948, 831)    |      10
  14 | (1018, 1011, 1007)                                     |       3
  15 | (1020, 1019, 1017, 1015, 1014, ..., 10, 9, 8, 6, 4, 3) |     912
  16 | (1021)                                                 |       1
  17 | (1022)                                                 |       1
  18 | (1023)                                                 |       1
A263327(5) = 16, A263327(16) = 12, A263327(12) = 5, so (5 16 12) = (16 12 5) is a 3-cycle. For all other cycles of length > 1, the order in which the terms occur under the map (e.g. 1018 -> 1007 -> 1011 -> 1018 for row 14) is different from the decreasing order given above. - _M. F. Hasler_, Dec 11 2019

Reinhard Zumkeller, Rows n = 1..18 of triangle, flattened, 1024 terms

Cf. A263327, A263383 (row lengths), A263329.

import Data.List ((\\), sort)
a263355 n k = a263355_tabf !! (n-1) !! (k-1)
a263355_row n = a263355_tabf !! (n-1)
a263355_tabf = sort $ cc a263327_list where
   cc [] = []
   cc (x:xs) = (reverse $ sort ys) : cc (xs \\ ys)
      where ys = x : c x
            c z = if y /= x then y : c y else []
                  where y = a263327 z

{M=0; (C(x,L=[x])=until(x==L[1], M+=1<A263327[x]));L); vecsort(vector(18,i,vecsort(C(valuation(M+1,2)),,12)))} \\ append [^15] to remove the long row 15. - M. F. Hasler, Dec 11 2019

Edited by M. F. Hasler, Dec 11 2019

A263383 Lengths of cycles of permutation A263327.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 10, 1, 74, 1, 1, 10, 3, 912, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Oct 16 2015

Cycle type of permutation A263327 = (1^12, 3^2, 10^2, 74, 912);
A263355(k,1) = A263329(k) for k such that a(k) = 1.
row lengths of A263355;
number of terms = 18; sum of terms = 1024.

			See A263355.

Cf. A263327, A263329, A263355.

Haskell
```
a263383 = length . a263355_row
```

cp's OEIS Frontend

A263327 A permutation of {0, 1, ..., 1023} corresponding to lexicographical ordering A262557 of numbers with decreasing digits A009995.

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A263328 Permutation of {0, ..., 1023} corresponding to lexicographic ordering of numbers with decreasing digits (A009995). Inverse of A263327.

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A263355 Table read by rows: cycles of the permutation A263327, sorted in increasing order of their largest element. The elements in each cycle are listed in decreasing numerical order.

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A263383 Lengths of cycles of permutation A263327.

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