### A Relationship Between Generalized Davenport-Schinzel Sequences and Interval Chains

#### Abstract

Let an $(r, s)$-formation be a concatenation of $s$ permutations of $r$ distinct letters, and let a block of a sequence be a subsequence of consecutive distinct letters. A $k$-chain on $[1, m]$ is a sequence of $k$ consecutive, disjoint, nonempty intervals of the form $[a_{0}, a_{1}] [a_{1}+1, a_{2}] \ldots [a_{k-1}+1, a_{k}]$ for integers $1 \leq a_{0} \leq a_{1} < \ldots < a_{k} \leq m$, and an $s$-tuple is a set of $s$ distinct integers. An $s$-tuple stabs an interval chain if each element of the $s$-tuple is in a different interval of the chain. Alon et al. (2008) observed similarities between bounds for interval chains and Davenport-Schinzel sequences, but did not identify the cause.

We show for all $r \geq 1$ and $1 \leq s \leq k \leq m$ that the maximum number of distinct letters in any sequence $S$ on $m+1$ blocks avoiding every $(r, s+1)$-formation such that every letter in $S$ occurs at least $k+1$ times is the same as the maximum size of a collection $X$ of (not necessarily distinct) $k$-chains on $[1, m]$ so that there do not exist $r$ elements of $X$ all stabbed by the same $s$-tuple.

Let $D_{s,k}(m)$ be the maximum number of distinct letters in any sequence which can be partitioned into $m$ blocks, has at least $k$ occurrences of every letter, and has no subsequence forming an alternation of length $s$. Nivasch (2010) proved that $D_{5, 2d+1}(m) = \Theta( m \alpha_{d}(m))$ for all fixed $d \geq 2$. We show that $D_{s+1, s}(m) = \binom{m- \lceil \frac{s}{2} \rceil}{\lfloor \frac{s}{2} \rfloor}$ for all $s \geq 2$. We also prove new lower bounds which imply that $D_{5, 6}(m) = \Theta(m \log \log m)$ and $D_{5, 2d+2}(m) = \Theta(m \alpha_{d}(m))$ for all fixed $d \geq 3$.

We show for all $r \geq 1$ and $1 \leq s \leq k \leq m$ that the maximum number of distinct letters in any sequence $S$ on $m+1$ blocks avoiding every $(r, s+1)$-formation such that every letter in $S$ occurs at least $k+1$ times is the same as the maximum size of a collection $X$ of (not necessarily distinct) $k$-chains on $[1, m]$ so that there do not exist $r$ elements of $X$ all stabbed by the same $s$-tuple.

Let $D_{s,k}(m)$ be the maximum number of distinct letters in any sequence which can be partitioned into $m$ blocks, has at least $k$ occurrences of every letter, and has no subsequence forming an alternation of length $s$. Nivasch (2010) proved that $D_{5, 2d+1}(m) = \Theta( m \alpha_{d}(m))$ for all fixed $d \geq 2$. We show that $D_{s+1, s}(m) = \binom{m- \lceil \frac{s}{2} \rceil}{\lfloor \frac{s}{2} \rfloor}$ for all $s \geq 2$. We also prove new lower bounds which imply that $D_{5, 6}(m) = \Theta(m \log \log m)$ and $D_{5, 2d+2}(m) = \Theta(m \alpha_{d}(m))$ for all fixed $d \geq 3$.

#### Keywords

alternations; formations; generalized Davenport-Schinzel sequences; interval chains; inverse Ackermann functions; permutations