. . "Davenport-Schinzel sequences are sequences that give rise to a hierarchy of functions that grow extremely close to linearly. An \\((n,s)\\) Davenport-Schinzel sequence is a sequence \\(a_1,a_2,\\ldots,a_m\\) where \\(a_i\\) are integers in \\([1,n]\\), such that: \n* \\(a_i \neq a_{i+1}\\) \n* There is no sequence \\(i_1 < i_2 < \\cdots < i_{s + 2}\\) such that \\(a_{i_1} = a_{i_3} = a_{i_5} = \\ldots \neq a_{i_2} = a_{i_4} = a_{i_6} = \\ldots\\)."@en . "Davenport-Schinzel sequence"@en . "Davenport-Schinzel sequences are sequences that give rise to a hierarchy of functions that grow extremely close to linearly. An \\((n,s)\\) Davenport-Schinzel sequence is a sequence \\(a_1,a_2,\\ldots,a_m\\) where \\(a_i\\) are integers in \\([1,n]\\), such that: \n* \\(a_i \neq a_{i+1}\\) \n* There is no sequence \\(i_1 < i_2 < \\cdots < i_{s + 2}\\) such that \\(a_{i_1} = a_{i_3} = a_{i_5} = \\ldots \neq a_{i_2} = a_{i_4} = a_{i_6} = \\ldots\\). Define \\(\\lambda_s(n)\\) as the length of the longest \\((n,s)\\) Davenport-Schinzel sequence. Then \\(\\lambda_1(n) = n\\), \\(\\lambda_2(n) = 2n - 1\\), \\(\\lambda_3(n) = \\Theta(n\\alpha(n))\\) where \\(\\alpha\\) is the inverse Ackermann function. A large number function can be extracted by defining \\(\text{DS}_3(n) = \\min\\{m : \\lambda_3(m) \\geq mn\\}\\)."@en .