Subfield: Combinatorics


Question: An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:

$$
\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
$$

Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write

$$
S_{3}[263415]=(132,312,231,213)
$$

More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.

In this power question, you will be asked to analyze some of the properties of labels and signatures.


We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :

![](https://cdn.mathpix.com/cropped/2023_12_21_793ce3b512229d118445g-1.jpg?height=236&width=442&top_left_y=2213&top_left_x=836)



A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:

![](https://cdn.mathpix.com/cropped/2023_12_21_54786ae6b210123098f4g-1.jpg?height=257&width=464&top_left_y=419&top_left_x=825)
Determine the smallest $p$ for which the 20-label


$$
L=3,11,8,4,17,7,15,19,6,2,14,1,10,16,5,12,20,9,13,18
$$

has a unique $p$-signature.

Solution:
The answer is $p=16$. To show this fact we will need to extend the idea from part 8(b) about "linking" inequalities forced by the various windows:

Theorem: A $p$-signature for an $n$-label $L$ is unique if and only if for every $k<n, k$ and $k+1$ are in at least one window together. That is, the distance between them in the $n$-label is less than $p$.

Proof: Suppose that for some $k$, the distance between $k$ and $k+1$ is $p$ or greater. Then the label $L^{\prime}$ obtained by swapping $k$ and $k+1$ has the same $p$-signature, because there are no numbers between $k$ and $k+1$ in any window and because the two numbers never appear in the same window.

If the distance between all such pairs is less than $k$, we need to show that $S_{p}[L]$ is unique. For $i=1,2, \ldots, n$, let $r_{i}$ denote the position where $i$ appears in $L$. For example, if $L=4123$, then $r_{1}=2, r_{2}=3, r_{3}=4$, and $r_{4}=1$.

Let $L=a_{1}, a_{2}, \ldots, a_{n}$. Since 1 and 2 are in some window together, $a_{r_{1}}<a_{r_{2}}$. Similarly, for any $k$, since $k$ and $k+1$ are in some window together, $a_{r_{k}}<a_{r_{k+1}}$. We then get a linked inequality $a_{r_{1}}<a_{r_{2}}<\cdots<a_{r_{n}}$, which can only be satisfied if $a_{r_{1}}=1, a_{r_{2}}=$ $2, \ldots, a_{r_{n}}=n$. Therefore, $S_{p}[L]$ is unique.

From the proof above, we know that the signature is unique if and only if every pair of consecutive integers coexists in at least one window. Therefore, we seek the largest distance between consecutive integers in $L$. That distance is 15 (from 8 to 9 , and from 17 to 18). Thus the smallest $p$ is 16 .

Final answer: 16