## 5.245. max_occ_of_consecutive_tuples_of_values

Origin

Design.

Constraint

$\mathrm{𝚖𝚊𝚡}_\mathrm{𝚘𝚌𝚌}_\mathrm{𝚘𝚏}_\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}\left(\mathrm{𝙼𝙰𝚇},𝙺,\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}\right)$

Type
 $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Arguments
 $\mathrm{𝙼𝙰𝚇}$ $\mathrm{𝚒𝚗𝚝}$ $𝙺$ $\mathrm{𝚒𝚗𝚝}$ $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚎𝚌}-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\right)$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚟𝚊𝚛}\right)$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|\ge 2$ $\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\right)$ $\mathrm{𝙼𝙰𝚇}\ge 1$ $𝙺\ge 2$ $𝙺<|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|\ge 1$ $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$
Purpose

$\mathrm{𝙼𝙰𝚇}$ is equal to the maximum number of occurrences of identical vectors derived from the vectors $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ in the following way. To each vector $〈{v}_{1},{v}_{2},\cdots ,{v}_{m}〉$ of $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ (with ${v}_{1},{v}_{2},\cdots ,{v}_{m}$ distinct) we generate all vectors $〈{u}_{1},{u}_{2},\cdots ,{u}_{𝙺}〉$ such that ${u}_{1}={v}_{p}$, ${u}_{2}={v}_{p+1}$, $\cdots$, ${u}_{𝙺}={v}_{p+𝙺-1}$ or ${u}_{1}={v}_{p+𝙺-1}$, ${u}_{2}={v}_{p+𝙺-2}$, $\cdots$, ${u}_{𝙺}={v}_{p}$ (with $1\le p\le m-𝙺+1$).

Example
$\left(1,2,〈\mathrm{𝚟𝚎𝚌}-〈4,1,3〉,\mathrm{𝚟𝚎𝚌}-〈2,7,6〉,\mathrm{𝚟𝚎𝚌}-〈5,9,8〉〉\right)$

Given the three vectors of the example we respectively generate:

• the pairs $〈4,1〉$, $〈1,4〉$, $〈1,3〉$, $〈3,1〉$ from the triple $〈4,1,3〉$,

• the pairs $〈2,7〉$, $〈7,2〉$, $〈7,6〉$, $〈6,7〉$ from the triple $〈2,7,6〉$,

• the pairs $〈5,9〉$, $〈9,5〉$, $〈9,8〉$, $〈8,9〉$ from the triple $〈5,9,8〉$.

Putting these pairs together, we get the set of pairs $\left\{〈1,3〉$, $〈1,4〉$, $〈2,7〉$, $〈3,1〉$, $〈4,1〉$, $〈5,9〉$, $〈6,7〉$, $〈7,2〉$, $〈7,6〉$, $〈8,9〉$, $〈9,5〉$, $〈9,8〉\right\}$. The $\mathrm{𝚖𝚊𝚡}_\mathrm{𝚘𝚌𝚌}_\mathrm{𝚘𝚏}_\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚞𝚙𝚕𝚎𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}$ constraint holds since the components of each of the original three vectors are distinct, and since $\mathrm{𝙼𝙰𝚇}$ is set to one and all the generated pairs are distinct.

Typical
 $\mathrm{𝙼𝙰𝚇}=1$ $𝙺=2$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|>2$
Arg. properties
• Functional dependency: $\mathrm{𝙼𝙰𝚇}$ determined by $𝙺$ and $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$.

• Contractible wrt. $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ when $\mathrm{𝙼𝙰𝚇}=1$.

Usage

This constraint occurs in balanced block design problems [RosaHuang71].