⚜ PRINCIPIA ORTHOGONA · Book XV · Ch 3← Ch 2 · What the Count Leaves Out · Book XV index
Book XV · Chapter 3 · counting model

Thirty Days

When a law tells every covered worker to take a vacation each year, it creates a demand for travel before anyone buys a ticket. This chapter counts what the law fixes and says what it leaves open.
The counts are exact for the rule as stated below. The rule comes from a secondary source, a law firm’s page; the statute text has not been read here, so check it before citing. No population, survey or tourism figure is used.

1 · The rule

As that source states it: after each 12-month qualifying period an employee in Brazil is entitled to paid vacation plus a one-third bonus (Constitution art. 7, XVII; CLT art. 134). The length depends on unjustified absences (CLT art. 130): up to 5 absences gives 30 calendar days, 6 to 14 gives 24, 15 to 23 gives 18, 24 to 32 gives 12, and above 32 gives none. Since Lei 13.467/2017 the vacation may be split into at most three periods, if the employee agrees; one period must be at least 14 calendar days and each of the others at least 5. The bonus is extra pay, not extra days, so it does not enter the counts.

2 · Counting the splits

Count ordered splits, since the order of the periods in the year matters to when a person travels. A vacation taken whole counts as one way and is not subject to the 14-day minimum, which applies when splitting.

CountsFor 30 days: 1 way unsplit; 21 ways in two periods (the first part runs from 5 to 25 days, and the larger part is always at least 15); 84 ways in three periods. Total: 106.

The three-period count has a closed form. With each part at least 5, subtract 5 from each and distribute 15 over three parts: 136 ways. Those with every part at most 13 (so no part reaches 14) number 52 by inclusion and exclusion, which leaves 136 − 52 = 84. The script confirms 84 by listing every split.

entitlement (days)unsplittwo periodsthree periodstotal
3012184106
241123 (14+5+5 in some order)16
181001
121001

So the rule leaves a worker with 30 days 106 legal ways to take it, and a worker who has missed enough days to fall to 18 or 12 only one: all in a single block.

3 · From entitlement to demand

A covered worker with a 30-day entitlement produces 30 person-days of vacation a year. If the vacations of covered workers were spread evenly over a 365-day year, then on any one day 30/365 = 0.0822 of them would be on vacation, about one in twelve. If instead every vacation fell inside one 30-day month, the share on vacation during that month would be 1, a factor of 365/30 = 12.17 over the even spread. That is an extreme case chosen for the arithmetic, not a forecast.

The law fixes the total and the shape of the pieces; it does not fix the dates. The dates decide the peak, and the peak is the demand d that the network of chapter 1 has to carry. This is where the book’s models meet: the count of covered workers and the dates they choose are inputs from outside the chapter.

What this does not show It does not say how many people hold an entitlement, when they take it, whether they travel, where they go or what they spend. It uses no population, survey, tourism or employment figure; the number of covered workers would need a source held and named on this page. It counts the legal splits of the rule as a secondary source states it, and does not cover exceptions, collective agreements or the sale of part of the vacation.

4 · Run it

python3 book15/ch03-verify.py uses only the standard library, prints every number on this page and exits non-zero on any failure. The Lean file for these counts is not written.