The ancient Maya people had two somewhat distinct calendar systems.

In somewhat simplified terms, one is a cyclical calendar known as The Calendar Round, that meshes several sacred and civil cycles; the other is an offset calendar known as The Long Count which is an offset calendar similar in many ways to the Gregorian calendar.

The Calendar Round

The Calendar Round has several intermeshing sacred and civil cycles that uniquely identify a specific date in an approximately 52 year cycle.

The Tzolk’in

The sacred cycle in the Mayan calendar round was called the Tzolk’in. The Tzolk'in has a cycle of 20 day names.

The twenty day names in the Maya Sacred Calendar are:

   Imix
   Ik’
   Ak’bal
   K’an
   Chikchan
   Kimi
   Manik’
   Lamat
   Muluk
   Ok
   Chuwen
   Eb
   Ben
   Hix
   Men
   Kib
   Kaban
   Etz’nab
   Kawak
   Ajaw

Intermeshed with the named days, the Tzolk’in has a cycle of 13 numbered days; 1 through 13. Every day has both a number and a name that repeat in a 260 day cycle.

For example:

   1 Imix
   2 Ik’
   3 Ak’bal
   ...
   11 Chuwen
   12 Eb
   13 Ben
   1 Hix
   2 Men
   3 Kib
   ... and so on.

The Haab

The Mayan civil calendar is called the Haab. This calendar has 365 days per year, and is sometimes called the ‘vague year.’ It is substantially the same as our year, but does not make leap year adjustments, so over long periods of time, gets out of synchronization with the seasons. It consists of 18 months with 20 days each, and the end of the year, a special month of only 5 days, giving a total of 365. The 5 days of the month of Wayeb (the last month), are usually considered to be a time of bad luck. This calendar is still used in parts of South America.

Each month in the Haab has a name. The Mayan names for the civil months are:

   Pop
   Wo
   Sip
   Sotz’
   Sek
   Xul
   Yaxk’in
   Mol
   Ch’en
   Yax
   Sak
   Keh
   Mak
   K’ank’in
   Muwan
   Pax
   K’ayab
   Kumk’u
   Wayeb (Short, "unlucky" month)

The months function very much like ours do. That is, for any given month we count through all the days of that month, and then move on to the next month.

Normally, the day 1 Pop is considered the first day of the civil year, just as 1 January is the first day of our year. In 2019, 1 Pop falls on April 2nd. But, because of the leap year in 2020, 1 Pop falls on April 1st in the years 2020-2023.

The only really unusual aspect of the Haab calendar is that, although there are 20 (or 5) days in each month, the last day of the month is not called the 20th (5th). Instead, the last day of the month is referred to as the ‘seating,’ or ‘putting in place,’ of the next month. (Much like how in our culture, December 24th is Christmas Eve and December 31st is 'New-Years Eve'.) In the language of the ancient Maya, the word for seating was chum, So you might have:

   ...
   18 Pop (18th day of the first month)
   19 Pop (19th day of the first month)
   Chum Wo (20th day of the first month)
   1 Wo (1st day of the second month)
   ... and so on.

Dates for any particular day are a combination of the Tzolk’in sacred date, and Haab civil date. When put together we get the “Calendar Round.”

Calendar Round dates always have two numbers and two names, and they are always written in the same order:

   (1) the day number in the Tzolk’in
   (2) the day name in the  Tzolk’in
   (3) the day of the month in the Haab
   (4) the month in the Haab

A calendar round is also a repeating cycle with a period of just short of 52 Gregorian calendar years. To be precise: it is 52 x 365 days. (No leap days)

Lords of the Night

A third cycle of nine days honored the nine Lords of the Night; nine deities that were associated with each day in turn. The names of the nine deities are lost; they are now commonly referred to as G1 through G9. The lord of the night may or may not be included in a Mayan date, if it is, it is typically just the appropriate G(x) at the end.

The Long Count

Mayans had a separate independent way of measuring time that did not run in cycles. (At least, not on normal human scales.) Instead, much like our yearly calendar, each day just gets a little further from the starting point. For the ancient Maya, the starting point was the ‘creation date’ of the current world. This date corresponds to our date of August 11, 3114 B.C. Dates are calculated by measuring how many days have transpired since this starting date; This is called “The Long Count.” Rather than just an absolute count of days, the long count is broken up into unit blocks, much like our calendar has months, years, decades and centuries.

   The basic unit is a k’in - one day.
   A 20 day month is a winal.
   18 winal (360 days) is a tun - sometimes referred to as a short year.
   20 short years (winal) is a k’atun
   20 k’atun is a bak’tun

A complete era is 13 bak’tun.

There are longer units too:

   Piktun == 20 Bak’tun (8,000 short years)
   Kalabtun  == 20 Piktun (160,000 short years)
   Kinchiltun  == 20 Kalabtuns (3,200,000 short years)

For the most part, the Maya only used the blocks of time up to the bak’tun. One bak’tun is around 394 years, much more than a human life span, so that was all they usually needed to describe dates in this era, or this world. It is worth noting, the two calendars working together allow much more accurate and reliable notation for dates than is available in many other calendar systems; mostly due to the pragmatic choice to make the calendar simply track days, rather than trying to make it align with seasons and/or try to conflate it with the notion of time.

Mayan Date correlations

There is some controversy over finding a correlation point between the Gregorian and Mayan calendars. The Gregorian calendar is full of jumps and skips to keep the calendar aligned with the seasons so is much more difficult to work with. There is some disagreement over which date the current Mayan cycle beginning correlates to in the Gregorian calendar. The most commonly used correlation factor is The GMT: 584283. Julian 584283 is a day count corresponding Mon, Aug 11, 3114 BCE in the Gregorian calendar, and the final day in the last Mayan long count cycle: 13.0.0.0.0 which is referred to as "the day of creation" in the Mayan calendar. There is nothing in known Mayan writing or history that suggests that a long count "cycle" resets every 13 bak’tun, judging by their other practices, it would make much more sense for it to reset at 20, if at all.

The reason reason there was much interest at all outside historical scholars in that the Mayan calendar is that the long count rolled over to 13.0.0.0.0 on Fri, Dec 21, 2012 (using the most common GMT correlation factor) prompting conspiracy theorists to predict a cataclysmic "end-of-the-word" scenario.

Excerpts from, and recommended reading:

From the website of the Foundation for the Advancement of Mesoamerican Studies, Inc.

Pitts, Mark. The complete Writing in Maya Glyphs Book 2 – Maya Numbers and the Maya Calendar. 2009. Retrieved 2019-01-19.
http://www.famsi.org/research/pitts/MayaGlyphsBook2.pdf

The task:

Write a reusable routine that takes a Gregorian date and returns the equivalent date in the Mayan in the Calendar Round and th e Long Count. At a minimum, use the GMT correlation. If desired, support other correlations. Using the GMT correlation, the following Gregorian and Mayan dates are equivalent:

Dec 21, 2012 (Gregorian), 4 Ajaw 3 K’ank’in G9 (Calendar round), 13.0.0.0.0 (Long count)

Support looking up dates for at least 50 years before and after the Mayan Long Count rollover: Dec 21, 2012. (Will require taking into account leap days.) Show the output here, on this page, for at least the following dates: (Note that these are in ISO 8601 format - yyyy-mm-dd. There is no requirement to use ISO 8601 format in your program, but if you differ, make a note of the expected format.)