Siteswap

Siteswap 531 state diagram

Not all siteswap sequences are valid.[9] All vanilla, synchronous, and multiplex siteswap sequences are valid if their state transitions create a cycle in their state diagram graph.[9] Sequences that do not create a cycle are invalid. For example, The pattern 531 can be mapped to a state diagram as shown on the right. Since the transitions in this sequence create a cycle in the graph, this pattern is valid.

There are other methods of determining a sequence's validity based on the flavor of siteswap.

A vanilla siteswap sequence ${\displaystyle a_{0}a_{1}a_{2}...a_{n-1}}$where ${\displaystyle n}$ is the period of the siteswap, is valid when the cardinality of the set ${\displaystyle S}$ (written in Set-builder notation) is equal to the period ${\displaystyle n}$ where

$${\displaystyle S=\{(a_{i}+i){\bmod {n}}|0\leq i\leq n-1\}}$$

To find if a pattern is valid, first create a new sequence formed by adding ${\displaystyle 0}$ to the first number, ${\displaystyle 1}$ to the second number, ${\displaystyle 2}$ to the third number and so on. Second, calculate the modulus (remainder) of each number with the period. If none of the numbers are duplicated in this final sequence, then the pattern is valid.[10]

For example, the pattern 531 would produce ${\displaystyle 5+0,3+1,1+2}$ or ${\displaystyle 5,4,3}$. Since the pattern 531 has a period of 3, The results from the previous example would produce ${\displaystyle 5{\bmod {3}},4{\bmod {3}},3{\bmod {3}}}$or ${\displaystyle 2,1,0}$. In this case, 531 is valid since the numbers ${\displaystyle 2,1,0}$ are all unique. Another example, 513 is an invalid pattern because the first step produces ${\displaystyle 5+0,1+1,3+2}$ or ${\displaystyle 5,2,5}$, the second step produces ${\displaystyle 5{\bmod {3}},2{\bmod {3}},5{\bmod {3}}}$ or ${\displaystyle 2,2,2}$, and the final sequence contains at least a duplicate of one number, in this case a 2.

A synchronous siteswap is valid if

1. it only contains even numbers and
2. it can be converted into a valid vanilla siteswap using the slide property.

otherwise it is invalid.

New valid vanilla sequences can be generated by swapping adjacent elements from another valid vanilla siteswap sequence, adding 1 to the number being swapped to the right and subtracting 1 from the number being swapped to the left.[10] The swap property will convert the valid sequence ${\textstyle a_{0}a_{1}a_{2}...a_{i}a_{i+1}...a_{n-1}}$ with arbitrary value ${\displaystyle i}$, to generate the new valid sequence ${\displaystyle a_{0}a_{1}a_{2}...(a_{i+1}+1)(a_{i}-1)...a_{n-1}}$.

For example, the swap property performed on the inner two throws of the sequence 4413 would move the 4 to the right subtracting 1 from it to become 3 and move the 1 to the left adding 1 to it to become 2. This produces the new valid siteswap pattern 4233.

A valid synchronous sequence can be converted to a valid asynchronous sequence and vice versa using the slide property. Given the synchronous sequence ${\displaystyle (a_{0},a_{1})(a_{2},a_{3})...(a_{n-2},a_{n-1})}$ the new vanilla sequences can be formed: ${\textstyle b_{0}b_{1}...b_{n-1}}$ where

$${\displaystyle b_{i}={\begin{cases}a_{i}+1,&{\text{if }}i{\text{ is even and }}a_{i}{\text{ crosses hands}}\\a_{i}-1,&{\text{if }}i{\text{ is odd and }}a_{i}{\text{ crosses hands}}\\a_{i},&{\text{otherwise}}\end{cases}}}$$

and ${\displaystyle c_{0}c_{1}...c_{n-1}}$ where

$${\displaystyle c_{i}={\begin{cases}a_{i+1}+1,&{\text{if }}i{\text{ is even and }}a_{i}{\text{ crosses hands}}\\a_{i-1}-1,&{\text{if }}i{\text{ is odd and }}a_{i}{\text{ crosses hands}}\\a_{i+1},&{\text{if }}i{\text{ is even and }}a_{i}{\text{ does not cross}}\\a_{i-1},&{\text{if }}i{\text{ is odd and }}a_{i}{\text{ does not cross}}\end{cases}}}$$

The slide property gets its name by sliding the throw times of one of the hands by one time unit so that the throws align asynchronously.[9]

For example, the siteswap (8x,4x)(4,4) would create two asynchronous (vanilla) siteswaps using the slide property: 9344 and 5744.

Siteswaps may be considered either prime or composite.[9] A siteswap is prime if the path created in its state diagram does not traverse any state more than once. Siteswaps that are not prime are called composite.

A non-rigorous but simpler method of determining if a siteswap is prime is to try to split it into any valid shorter pattern which uses the same number of props.[9] For example, 44404413 can be split into 4440, 441, and 3; therefore, 44404413 is composite. Another example, 441, which uses three props, is prime, as 1, 4, 41, and 44 are not valid three prop patterns (as 1/3≠3, 4/3≠3, (4+1)/3≠3, and (4+4)/3≠3). Sometimes this process does not work; for example, 153 (better known by its rotation 531) looks like it can be split into 15 and 3, but checking that the cycle has no repeating nodes in the graph traversal indicates that it is prime by the more rigorous definition.

It has been shown empirically that the longest prime siteswaps bounded by height ${\displaystyle h}$ contain mostly the throws ${\displaystyle 0}$ and ${\displaystyle h}$.[11] The longest prime patterns with height 22 (with 3 ball maximum), for 9 balls (with 13 maximum height), and for heights and ball counts in between, were enumerated by Jack Boyce in February 1999 using a program called jdeep.[12] The full list of longest prime siteswaps generated by jdeep (with '0' throws represented by a '-' and maximum height throws represented by a '+') can be found here.

Vanilla siteswap patterns may be viewed as certain elements of the affine Weyl group of type ${\displaystyle {\tilde {A}}_{n}}$.[13] One presentation of this group is as the set of bijective functions f on the integers such that, for a fixed n: f(i + n) = f(i) + n for all integers i. If the element f satisfies the further condition that f(i) ≥ i for all i, then f corresponds to the (infinitely repeated) siteswap pattern whose ith number is f(i) − i: that is, the ball thrown at time i will land at time f(i).

A subset of these siteswap patterns naturally label strata in the positroid stratification of the Grassmannian.[14]