1,3

There are a(1) 1-positional system 1!=1 numbers

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

There are a(4)-a(9) 3-positional system 3!=6 numbers

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

There are a(34)-a(153) 5-positional system 5!=120 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.

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

a(3)=5 because smallest number in trinary system with 3 different digits is 012=5

a(4)=7 second number in trinary system with 3 different digits is 021=7

a(5)=11 third number in trinary system with 3 different digits is 102=11

a(6)=15 120=15

etc.

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*)

Cf. A003422, A007489.

Sequence in context: A099835 A133436 A001225 this_sequence A032616 A006066 A084935

Adjacent sequences: A134637 A134638 A134639 this_sequence A134641 A134642 A134643

nonn

Artur Jasinski (grafix(AT)csl.pl), Nov 05 2007, Nov 07 2007, Nov 08 2007