A134640 - OEIS

%I #22 Oct 21 2022 07:01:27

%S 0,1,2,5,7,11,15,19,21,27,30,39,45,54,57,75,78,99,108,114,120,135,141,

%T 147,156,177,180,198,201,210,216,225,228,194,198,214,222,238,242,294,

%U 298,334,346,358,366,414,422,434,446,482,486,538,542,558,566,582,586

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

%C Note that leading zeros are allowed in these numbers.

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

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

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

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

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

%C ...

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

%C 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.

%C T(n,1) = A023811(n); T(n,A000142(n)) = A062813(n). - _Reinhard Zumkeller_, Aug 29 2014

%H Reinhard Zumkeller, <a href="/A134640/b134640.txt">Rows n = 1..7 of triangle, flattened</a>

%e We build permutational numbers:

%e a(1)=0 in unitary positional system we have only one digit 0

%e a(2)=1 because in binary positional system smaller number with two different digits is 01 = 1

%e a(3)=2 because in binary positional system bigger number with two different digits is 10 = 2 (binary system is over)

%e a(4)=5 because smallest number in ternary system with 3 different digits is 012=5

%e a(5)=7 second number in ternary system with 3 different digits is 021=7

%e a(6)=11 third number in ternary system with 3 different digits is 102=11

%e a(7)=15 120=15

%e etc.

%t 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*)

%t Flatten[Table[FromDigits[#,n]&/@Permutations[Range[0,n-1]],{n,5}]] (* _Harvey P. Dale_, Dec 09 2014 *)

%o (Haskell)

%o import Data.List (permutations, sort)

%o a134640 n k = a134640_tabf !! (n-1) !! (k-1)

%o a134640_row n = sort $

%o    map (foldr (\dig val -> val * n + dig) 0) $ permutations [0 .. n - 1]

%o a134640_tabf = map a134640_row [1..]

%o a134640_list = concat a134640_tabf

%o -- _Reinhard Zumkeller_, Aug 29 2014

%o (Python)

%o from itertools import permutations

%o def fd(d, b): return sum(di*b**i for i, di in enumerate(d[::-1]))

%o def row(n): return [fd(d, n) for d in permutations(range(n))]

%o print([an for r in range(1, 6) for an in row(r)]) # _Michael S. Branicky_, Oct 21 2022

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

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

%K nonn,base,tabf

%O 1,3

%A _Artur Jasinski_, Nov 05 2007, Nov 07 2007, Nov 08 2007

%E Corrected indices in examples. Replaced dashes in comments by the word "to" - _R. J. Mathar_, Aug 26 2009