# Lyndon words over $GF(4)$ 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 $GF(4)$ 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., $s=\sum_{1\leq i\lt j\leq n} a_i a_j$. Below we use $a=\operatorname{root}(x^2+x+1)$ and $b=1+a$.

1 2 3 4 5 6 (trace,subtrace) $n$ (0,0) (0,1)(0,$a$)(0,$b$) (1,0)($a$,0)($b$,0) (1,1)($a$,$b$)($b$,$a$) (1,$a$)(1,$b$)($a$,1)($a$,$a$)($b$,1)($b$,$b$) 1 0 1 0 0 0 0 1 1 0 2 1 1 2 1 6 2 4 4 4 15 12 15 12 12 40 40 45 45 40 153 144 144 153 144 528 496 512 512 512 1841 1813 1841 1813 1813 6528 6528 6579 6579 6528 23901 23808 23808 23901 23808 87550 87210 87380 87380 87380 322875 322560 322875 322560 322560 1198080 1198080 1198665 1198665 1198080 4474738 4473647 4473647 4474738 4473647

## Examples

The one 4-ary Lyndon word of trace $b$, subtrace $a$ and length 2 is $\{1a\}$. The two 4-ary Lyndon words of trace 0, subtrace 0 and length 3 are $\{123, 132\}$. The two 4-ary Lyndon words of trace 0, subtrace 1 and length 4 are $\{0011, 11aa\}$.