Here we consider the set S(n;t,s) of length n 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 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 .

Examples:

• The one ternary string of trace 2, subtrace 1 and length 2 is { 11 }.
• The three ternary strings of trace 0, subtrace 0 and length 3 are { 000, 111, 222 }.
• The three ternay strings of trace 1, subtrace 2 and length 3 are { 112, 121, 211 }.

Further Notes:

• The number S(n;t,s) can be computed from the following recurrence relation

  S(n;t,s) = S(n-1;t,s) + S(n-1;t-1,s-(t-1)) + S(n-1;t-2,s-2(t-2))

  Note that all operations involving operands t or s are carried out over GF(3).

• Column (0,0) is sequence A073947 in Neil J. Sloane's database of integer sequences.
• Column (0,1) is sequence A073948 in Neil J. Sloane's database of integer sequences.
• Column (0,2) is sequence A073949 in Neil J. Sloane's database of integer sequences.
• Column (1,0),(2,0) is sequence A073950 in Neil J. Sloane's database of integer sequences.
• Column (1,1),(2,1) is sequence A073951 in Neil J. Sloane's database of integer sequences.
• Column (1,2),(2,2) is sequence A073952 in Neil J. Sloane's database of integer sequences.

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