A134640 Permutational numbers (numbers with k different digits in k-positional system).

0, 1, 2, 5, 7, 11, 15, 19, 21, 27, 30, 39, 45, 54, 57, 75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, 210, 216, 225, 228, 194, 198, 214, 222, 238, 242, 294, 298, 334, 346, 358, 366, 414, 422, 434, 446, 482, 486, 538, 542, 558, 566, 582, 586

Offset

1,3

Comments

Note that leading zeros are allowed in these numbers.

a(1) is the 1-positional system 1!=1 numbers

a(2) to a(3) are two=2! 2-positional system numbers

a(4) to a(9) are six=3! 3-positional system numbers

a(10) to a(33) are 24=4! 4-positional system numbers

a(34) to a(153) are 120=5! 5-positional system numbers

...

There are a(!k)-a(Sum[m!,1,k])=a(A003422)-a(A007489) k-positional system k! numbers

The name permutational numbers arises because each permutation of k elements is isomorphic with one and only one of member of this sequence and conversely each number in this sequence is isomorphic with one and only one permutation of k elelmnts or its equivalent permutation matrix.

T(n,1) = A023811(n); T(n,A000142(n)) = A062813(n). - Reinhard Zumkeller, Aug 29 2014

Links

Reinhard Zumkeller, Rows n = 1..7 of triangle, flattened

Example

We build permutational numbers:
a(1)=0 in unitary positional system we have only one digit 0
a(2)=1 because in binary positional system smaller number with two different digits is 01 = 1
a(3)=2 because in binary positional system bigger number with two different digits is 10 = 2 (binary system is over)
a(4)=5 because smallest number in ternary system with 3 different digits is 012=5
a(5)=7 second number in ternary system with 3 different digits is 021=7
a(6)=11 third number in ternary system with 3 different digits is 102=11
a(7)=15 120=15
etc.

Mathematica

a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[w[[m]], n + 1]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 5}]; a (*Artur Jasinski*)
Flatten[Table[FromDigits[#, n]&/@Permutations[Range[0, n-1]], {n, 5}]] (* Harvey P. Dale, Dec 09 2014 *)

Prog

(Haskell)
import Data.List (permutations, sort)
a134640 n k = a134640_tabf !! (n-1) !! (k-1)
a134640_row n = sort $
   map (foldr (\dig val -> val * n + dig) 0) $ permutations [0 .. n - 1]
a134640_tabf = map a134640_row [1..]
a134640_list = concat a134640_tabf
-- Reinhard Zumkeller, Aug 29 2014
(Python)
from itertools import permutations
def fd(d, b): return sum(di*b**i for i, di in enumerate(d[::-1]))
def row(n): return [fd(d, n) for d in permutations(range(n))]
print([an for r in range(1, 6) for an in row(r)]) # Michael S. Branicky, Oct 21 2022

Crossrefs

Cf. A003422, A007489, A061845, A000142 (row lengths excluding 1st term).

Cf. A023811, A062813, A000142 (row lengths), A007489 (sums of row lengths).

Sequence in context: A001225 A372621 A157001 * A216094 A184857 A032616

Adjacent sequences: A134637 A134638 A134639 * A134641 A134642 A134643

Keyword

nonn,base,tabf

Author

Artur Jasinski, Nov 05 2007, Nov 07 2007, Nov 08 2007

Extensions

Corrected indices in examples. Replaced dashes in comments by the word "to" - R. J. Mathar, Aug 26 2009

Status

approved