Formations and generalized Davenport-Schinzel sequences
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An (r, s)-formation is a concatenation of s permutations of r distinct letters. We define the function F_r, s(n) to be the maximum possible length of a sequence with n distinct letters that avoids all (r, s)-formations and has every r consecutive letters distinct, and we define the function F_r, s(n, m) to be the maximum possible length of a sequence with n distinct letters that avoids all (r, s)-formations and can be partitioned into m blocks of distinct letters. (Nivasch, 2010) and (Pettie, 2015) found bounds on F_r, s(n) for all fixed r, s > 0, but no exact values were known, even for s = 2. We prove that F_r,2(n, m) = n+(r-1)(m-1), F_r,3(n, m) = 2n+(r-1)(m-2), F_r,2(n) = (n-r)r+2r-1, and F_r,3(n) = 2(n-r)r+3r-1, improving on bounds of (Klazar, 1992), (Nivasch, 2010), and (Pettie, 2015). In addition, we improve an upper bound of (Klazar, 2002). For any sequence u, define Ex(u, n) to be the maximum possible length of a sequence with n distinct letters that avoids u and has every r consecutive letters distinct, where r is the number of distinct letters in u. Klazar proved that Ex((a_1 ... a_r)^2, n) < (2n+1)L, where L = Ex((a_1 ... a_r)^2,K-1)+1 and K = (r-1)^4 + 1. Here we prove that K = (r-1)^4 + 1 in Klazar's bound can be replaced with K = (r-1)^3+1. We also prove a conjecture from (Geneson et al., 2014) by showing for t ≥ 1 that Ex(a b c (a c b)^t a b c, n) = n 2^1/t!α(n)^t± O(α(n)^t-1). In addition, we prove that Ex(a b c a c b (a b c)^t a c b, n) = n 2^1/(t+1)!α(n)^t+1± O(α(n)^t) for t ≥ 1. Furthermore, we extend the equalities F_r,2(n, m) = n+(r-1)(m-1) and F_r,3(n, m) = 2n+(r-1)(m-2) to formations in d-dimensional 0-1 matrices, sharpening a bound from (Geneson, 2019).
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