Re: VMS words and Roman numerals

[Jorge Stolfi <stolfi@xxxxxxxxxxxxx>]

There's more: the Roman numerals with the *subtractive* notation 
(IV = 4, IX = 9, etc.) have a symmetric digit length distribution, too.
Let again d_k be the number of distinct Roman digits with given length k:

  k  d_k  digits
  -  ---  -------
  0   1   (empty)
  1   2   I V
  2   4   II IV VI IX
  3   2   III VII 
  4   1   VIII
  
In fact, the distribution d_k is not too different from binom(4,k) =
1:4:6:4:1 times a constant. Therefore the subtractive Roman numerals
have a symmetric length distribution, too. Let again R_k be the number
of distinct Roman numerals from 0 to 999 with given length k. Here are
the values:

    k   R_k  numerals
   ---  ---  ------------
    0     1  (empty)
    1     6  I V X L C D 
    2    24  II IV VI IX XI XV XX XL XC LI ... CX CL CD CM
    3    62
    4   123
    5   180
    6   208
    7   180
    8   123
    9    62
   10    24
   11     6  
   12     1  DCCCLXXXVIII

The distribution R_k is actually quite close to binom(12,k):

    k   R_k  binm  ratio
   ---  ---  ----  --------
    0     1     1  1.00
    1     6    12  0.50
    2    24    66  0.36
    3    62   220  0.28
    4   123   495  0.25
    5   180   792  0.23
    6   208   924  0.23
    7   180   792  0.23
    8   123   495  0.25
    9    62   220  0.28
   10    24    66  0.36
   11     6    12  0.50
   12     1     1  1.00

Note that the ratio R_k/binom(12,k) is close to 0.23 over most of the
significant range.

All the best,

--stolfi