a pleasing notation

[Antti Karttunen]

Marc LeBrun wrote:
> 
> Briefly, I've lately been writing Zb[n](x) for the polynomial in argument x
> whose coefficients are the corresponding digits of index n in base b.
> 
> It facilitates expressing many identities, including natural but
> notationally neglected concrete "numerical" operations (digit reversal, for
> instance) and overall just seems to promote a useful perspicuity.
> 
> For example, "Z2[(5*4^k-2)/3](1/tau) = 1" might form a nice commentary on
> A020989.
> 
> It also suggests many interesting generalizations.  Perhaps you too will
> like this?

Yes, it has its merits, although the notation is a bit confusing
at first (one thinks Z2[x] polynomials first).
What about similar notations for the Zeckendorf expansion (Fibonacci
number system), and the factorial base notation?
  http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007623
(Would ZZ[...] or ZF[...] be a good notation for the former, and Z![...]
for the latter?)

When working with such infinite sequences of permutations as A055089:
  http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A055089
it seems that there are lot of hidden identities that could
be represented by considering the factorial expansion of n.

E.g. here's one conjecture which should be easily proved
(if one unravels the permutation generation algorithm a bit):
for each n, A059437[n] = A059437[A056019[n]] ???
That is, for each permutation's and its inverse's position
in the "reversed lexicographic sequence" A055089 the sum
of the digits in their factorial expansion are equal.

The sequence A059437
  http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A059437
is the sum of digits when n is written in factorial base,
and is essentially same as A034968, which just lacks the initial term 0.



Terveisin,

Antti Karttunen