# Lyndon words over $GF(2)$ with given trace and subtrace

Here we consider the number $L(n;t,s)$ of length $n$ Lyndon words $a_1 a_2 \ldots a_n$ over the alphabet consisting of the elements of the field $GF(2)$ that have trace $t$ and subtrace $s$. The trace of a Lyndon word is the sum of its digits over the field, i.e., $t = a_1 + a_2 + \cdots + a_n$. The subtrace is the sum of the products of all $n(n-1)/2$ pairs of digits taken over the field, i.e., $s=\sum_{1\leq i\lt j\leq n} a_i a_j$. Note that $GF(2)=\mathbb{Z}_2$.

1 2 3 4 5 (trace,subtrace) $n$ (0,0) (0,1) (1,0) (1,1) 1 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 1 2 1 2 2 2 2 3 5 4 4 5 8 6 8 8 15 13 15 13 24 24 27 24 45 48 48 45 80 85 85 85 155 160 155 160 288 288 288 297 550 541 541 550 1024 1008 1024 1024 1935 1920 1935 1920 3626 3626 3654 3626 6885 6912 6912 6885 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 $\{001111, 010111\}$.