Lyndon words over F₃ of given trace and subtrace.

Here we consider the set L(n;t,s) of length n lyndon words a₁a₂···a_n over the alphabet consisting of the elements of the field F₃ that have trace t and subtrace s. The trace of a lyndon word is the sum of its digits over the field F₃ , i.e. t = a₁+a₂+ ··· +a_n. The subtrace is the sum of the products of all n(n-1)/2 pairs of digits taken over the field F₃ , i.e. s = SUM( a_ia_j : 1 < i < j < n ). Note that F₃ = Z₃ .

	(trace,subtrace)
n	(0,0)	(0,1)	(0,2)	(1,0) (2,0)	(1,1) (2,1)	(1,2) (2,2)
1	1	0	0	1	0	0
2	0	0	1	1	0	0
3	0	0	2	1	1	1
4	2	1	3	2	1	3
5	4	6	6	6	4	6
6	9	14	15	13	13	13
7	32	36	36	32	36	36
8	90	93	87	87	90	93
9	240	252	234	243	243	243
10	654	661	645	654	661	645
11	1804	1782	1782	1782	1804	1782
12	4950	4893	4893	4914	4914	4914
13	13664	13608	13608	13664	13608	13608
14	37944	37890	37994	37994	37944	37890
15	106272	106142	106434	106288	106288	106288
16	298890	298755	299025	298890	298755	299025
17	843796	844182	844182	844182	843796	844182
18	2390595	2391732	2391723	2391363	2391363	2391363
19	6796160	6797196	6797196	6796160	6797196	6797196
20	19370696	19371684	19369708	19369708	19370696	19371684

Examples:

The two ternary Lyndon words of trace 0, subtrace 0 and length 4 are { 0111, 0222 }.
The three ternary Lyndon words of trace 1, subtrace 2 and length 4 are { 0112, 0121, 0211 }.
The six ternary Lyndon words of trace 2, subtrace 2 and length 5 are { 00122, 00212, 00221, 01022, 01202, 02021 }.

Further Notes:

L₃(n,0,0) is sequence A053548 in Neil J. Sloane's database of integer sequences.
L₃(n,0,1) is sequence A053560 in Neil J. Sloane's database of integer sequences.
L₃(n,0,2) is sequence A053561 in Neil J. Sloane's database of integer sequences.
L₃(n,2,0) = L₃(n,1,0) is sequence A053562 in Neil J. Sloane's database of integer sequences.
L₃(n,1,1) = L₃(n,2,1) is sequence A053563 in Neil J. Sloane's database of integer sequences.
L₃(n,1,2) = L₃(n,2,2) is sequence A053564 in Neil J. Sloane's database of integer sequences.

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Lyndon words over F3 of given trace and subtrace.

Examples:

Further Notes:

Lyndon words over F₃ of given trace and subtrace.