# Re: VMS words and Roman numerals

```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

```