Numbers in combinatorics

This page contains numbers appearing in combinatorics, which don’t fit on other lists.
== List of numbers appearing in combinatorics ==
*In the , there are 217 combinations of and day in the month, being the most (in)famous.
*Article 79 of the requires constitutional amendments to be approved by an absolute two-thirds majority of the Bundestag along with a simple two-thirds majority of the . Article 51 of the same law gives each state at least three votes, a fourth vote for states with more than 2 million inhabitants, a fifth vote for states with more than 6 million inhabitants, and a sixth vote for states with more than 7 million inhabitants. There are currently four states with three votes, seven states with four votes, with five votes, and four states with six votes. A calculation reveals that of the 65,536 possible voting patterns, 7,228 lead to an absolute two-thirds majority.
*The number 7,825 is the smallest n for which the {1, 2, 3, … , n} cannot be written as a of two , such that both of them contain no s.
*There are 17,152 solutions to the Ostomachion of Archimedes. However, this value is disputed, as it allows reflections of pieces and is just for square solutions, and not rectangles.https://www.nytimes.com/2003/12/14/us/in-archimedes-puzzle-a-new-eureka-moment.html
*3,628,800 is not only 10!, but is also the number of seconds in six weeks.
*The character has 11,007 possible mouth expressions and 4,429 possible brow expressions, and therefore 11,007×4,429 = 48,750,003 possible facial expressions.
*A 2011 paper by Coward and LackenbyAn upper bound on Reidemeister moves (p.4) (accessed 2021-03-07) provides an upper bound on the number of required to pass between two knots with the same number of crossings. The upper bound given is \(2 \uparrow \uparrow (2^{163 \times 2^{14} \times n}) \approx 2 \uparrow \uparrow (10^{803,928n})\) (where n is the number of crossings), which they simplify to the larger bound \(2 \uparrow \uparrow (10^{1,000,000n})\). This is one of the largest numbers ever used in a mathematical proof.
== Sudoku-related numbers ==
*405 is the sum of all the numbers on a 9×9 Sudoku grid. It is equal to 9×T9, where Tn is the nth . This number is also called Ternary-dust mite.
*6,670,903,752,021,072,936,960 is a combinatoric number equal to the number of possible 9×9 Sudoku grids.Mathematics of Sudoku - Wikipedia
**6,670,903,752,021,072,936,960 is roughly equal to \(f_2(66)\) in the fast-growing hierarchy.
*109,110,688,415,571,316,480,344,899,355,894,085,582,848,000,000,000 is the product of all the numbers on a 9×9 Sudoku grid. It is equal to 19×29×39×49×59×69×79×89×99, the product of the first 9 9th powers.
== Sources ==
Category:Numbers
Category:Lists
Category:Class 1
Category:Class 2
Category:Numbers with 3 digits
Category:Unnamed numbers with 4 to 6 digits
Category:Numbers with 7 to 29 digits
Category:Numbers with 30 to 100 digits