Binary Lyndon words with given trace and subtrace
A $k$ary
Lyndon word is a string made from the characters $\{0,1,\ldots,k1\}$.
It must be aperiodic (not equal to any of its nontrivial rotations) and be lexicographically least among its rotations.
Here we consider the number $L(n;t,s)$ of length $n$ Lyndon words $a_1 a_2 \ldots a_n$ over the alphabet $\{0,1,\ldots,k1\}$ with $k=2$ that have trace $t$ and subtrace $s$.
The
trace of a Lyndon word is the sum of its digits mod $k$, i.e., $t = a_1 + a_2 + \cdots + a_n \pmod{k}$.
The
subtrace is the sum of the products of all $n(n1)/2$ pairs of digits taken mod $k$, i.e., $s=\sum_{1\leq i\lt j\leq n} a_i a_j$.
 (trace,subtrace) 
$n$
 (0,0)
 (0,1)
 (1,0)
 (1,1)

1 
1  0  1
 0


2 
0  0  1
 0


3 
0  1  1
 0


4 
0  1  1
 1


5 
1  2  1
 2


6 
2  2  2
 3


7 
5  4  4
 5


8 
8  6  8
 8


9 
15  13  15
 13


10 
24  24  27
 24


11 
45  48  48
 45


12 
80  85  85
 85


13 
155  160  155
 160


14 
288  288  288
 297


15 
550  541  541
 550


16 
1024  1008  1024
 1024


17 
1935  1920  1935
 1920


18 
3626  3626  3654
 3626


19 
6885  6912  6912
 6885


20 
13056  13107  13107
 13107


Examples
The one binary Lyndon word of trace 1, subtrace 0 and length 3 is $\{001\}$.
The two binary Lyndon words of trace 0, subtrace 1 and length 5 are $\{00011, 00101\}$.
The two binary Lyndon words of trace 0, subtrace 0 and length 6 are $\{001111, 010111\}$.
Enumeration (OEIS)