The Mathematics of Juggling

Yuki Takahashi

University of California, Irvine

November 12, 2015

Co-sponcered by UCI Illuminations and Juggle Buddies

Yuki Takahashi (UC Irvine) November 12, 2015 1 / 51 Introduction: What is Juggling?

Figure : 2013, winter, 5-ball cascade

Juggling is the manipulation of objects (balls, clubs, rings, hats, cigar boxes, diabolos, devil sticks, yoyos, etc).

Yuki Takahashi (UC Irvine) November 12, 2015 2 / 51 Club Juggling

Figure : Korynn Aguilar (Chair), April 2015, UC Irvine, One World Concert.

Yuki Takahashi (UC Irvine) November 12, 2015 3 / 51 Passing

Figure : October 2015, University Hills, Fall Fiesta.

Yuki Takahashi (UC Irvine) November 12, 2015 4 / 51 (Dragon) Ball Juggling

Figure : Vegeta, juggling 7 (dragon) balls

Yuki Takahashi (UC Irvine) November 12, 2015 5 / 51 Papers about Siteswaps

A. Engsr¨om,L. Leskel¨a,H. Varpanen, Geometric juggling with q-analogues, Discrete Math. 338 (2015), 1067–1074. A. Ayyer, B, Arvind, S. Corteel, F. Nunzi, Multivariate juggling probabilities, Electron. J. Probab. 20 (2015). A. Knutson, T. Lam, D. Speyer, Positroid varieties: Juggling and geometry, Compos. Math. 149 (2013) 1710–1752. C. Elsner, D. Klyve, E. Tou, A zeta function for juggling sequences, J. Comb. Number Theory 4 (2012) 53–65. S. Butler, R. Graham, Enumerating (multiplex) juggling sequences, Ann. Comb. 13 (2010) 413–424. F. Chung, R. Graham, Primitive juggling sequences, Amer. Math. Monthly 115 (2008) 185–194.

Yuki Takahashi (UC Irvine) November 12, 2015 6 / 51 Juggling Patterns

Deﬁnition (Simple Juggling Patterns) In this talk, we assume the balls are juggled in a constant beat, that is, the throws occur at discrete equally spaced moments in time, patterns are periodic, and at most one ball gets caught and thrown at every beat.

Figure : Juggling diagram of the ?-ball cascade.

Yuki Takahashi (UC Irvine) November 12, 2015 7 / 51 Example (1)

Figure : Juggling diagram of the 4-ball fountain.

This pattern is denoted by 4.

Yuki Takahashi (UC Irvine) November 12, 2015 8 / 51 Example (2)

Figure : Juggling diagram of the 3-ball shower.

This pattern is denoted by ...?

Yuki Takahashi (UC Irvine) November 12, 2015 9 / 51 Example (3)

Figure : 4-ball shower.

This pattern is denoted by 71.

Yuki Takahashi (UC Irvine) November 12, 2015 10 / 51 Example (4)

This pattern is denoted by ...? The number of ball juggled is ...?

Yuki Takahashi (UC Irvine) November 12, 2015 11 / 51 Terminologies

We call a ﬁnite sequence of non-negative integers arising from a juggling pattern a juggling sequence. (Example: 3, 4, 51, 441, 7, 51515151, are all juggling sequences)

The length of a ﬁnite sequence of integers is called its period. (Example: 441441 has period 6)

A juggling sequence is minimal if it has minimal period among all the juggling sequences representing the same juggling pattern. (Example: 144 is minimal, but 441441 is not)

Yuki Takahashi (UC Irvine) November 12, 2015 12 / 51 Example (4)

546 is NOT a juggling sequence.

Figure : What is wrong...?

In general, if s = n(n − 1) ··· , then s is not a juggling sequence.

Yuki Takahashi (UC Irvine) November 12, 2015 13 / 51 The Average Theorem

Theorem (The Average Theorem) The number of balls necessary to juggle a juggling sequence is its average. If the average is not an integer, then the sequence is not a juggling sequence.

Example The number of balls necessary to juggle 441 is 3. 5+6+2 562 is NOT a juggling sequence (note that 3 = 4.3333... is not an integer).

Yuki Takahashi (UC Irvine) November 12, 2015 14 / 51 Swapping

i j i j

Figure : a juggling sequence s is transformed into another juggling sequence si,j .

This operation is called swapping.

Yuki Takahashi (UC Irvine) November 12, 2015 15 / 51 Example (1)

Let s = 441. Then s0,2 = ...?

Figure : s to s0,2.

Yuki Takahashi (UC Irvine) November 12, 2015 16 / 51 Example (1)

Figure : 441 to 342.

Yuki Takahashi (UC Irvine) November 12, 2015 17 / 51 Example (2)

Let s = 7531. Then s1,3 = ...?

Figure : s to s1,3.

Yuki Takahashi (UC Irvine) November 12, 2015 18 / 51 Example (3)

Let s = 6313. Then s1,2 = 6223.

Figure : s to s1,2.

Yuki Takahashi (UC Irvine) November 12, 2015 19 / 51 Swapping

Deﬁnition (Swapping) p−1 Let s = {ak }k=0 be a sequence of nonnegative integers. Let i and j be integers such that 0 ≤ i < j ≤ p − 1 and j − i ≤ ai . Let si,j be  a + j − i if k = i  j si,j (k) = ai − (j − i) if k = j  ak o.w.

i j i j

Yuki Takahashi (UC Irvine) November 12, 2015 20 / 51 Summary of Swapping

For a juggling sequence s, we denote the number of balls necessary to juggle it by ball(s). Then

The sequence s is a juggling sequence if and only if si,j is a juggling sequence.

The average of s is the same as the average of si,j .

If s is the juggling sequence, then ball(s) = ball(si,j ).

Yuki Takahashi (UC Irvine) November 12, 2015 21 / 51 Leeeeet’s Practice!!! =]

Example (Swapping)

Let s = 642. Then s0,1 = 552, and s0,2 = ...? (What does this result tell you about s? Is it a juggling sequence?)

Let s = 532. Then s0,1 = ...?

Let s = 123456789. Then s3,5 = ...?

Reminder: p−1 Let s = {ak }k=0 be a sequence of nonnegative integers. Then  a + j − i if k = i  j si,j (k) = ai − (j − i) if k = j  ak o.w.

Yuki Takahashi (UC Irvine) November 12, 2015 22 / 51 Getting Sleeply? It’s Show Time! :)

Yuki Takahashi (UC Irvine) November 12, 2015 23 / 51 Cyclic Shifts

Deﬁnition (Cyclic Shifts) p−1 Let s = {sk }k=0 be a sequence of nonnegative integers. Let

ˆs = a1a2a3 ··· ap−1a0.

We call the operation of transforming s into ˆs the cyclic shift of s.

Example: If s = 12345, then ˆs = 23451.

Yuki Takahashi (UC Irvine) November 12, 2015 24 / 51 Example

Let s = 441. Then ˆs = 414.

Figure : 441 to 414.

Yuki Takahashi (UC Irvine) November 12, 2015 25 / 51 Summery of Cyclic Shifts

Apparantly, we have the following: The sequence s is a juggling sequence if and only if ˆs is a juggling sequence. The average of s is the same as the average of ˆs. If s is a juggling sequence, then ball(s) = ball(ˆs).

Yuki Takahashi (UC Irvine) November 12, 2015 26 / 51 Summery!!!

Swapping and cyclic shifts both preserve

1 ”jugglibility”, 2 the average, and 3 the number of balls (if it is a juggling sequence).

Yuki Takahashi (UC Irvine) November 12, 2015 27 / 51 Flattening Algorithm

The ﬂattening algorithm takes as input an arbitrary sequence s.

1. If s is a constant sequence, stop and output this sequence. Otherwise,

2. use cyclic shifts to arrange the elements of s such that one of maximum height, say e, comes to rest at beat 0 and one not of maximum height, say f , comes to rest at beat 1. If e and f diﬀer by 1, stop and output this new sequence. Otherwise,

3. perform a swapping of beats 0 and 1, and return to step 1.

Yuki Takahashi (UC Irvine) November 12, 2015 28 / 51 Examples! =)

Denote the map s 7→ s0,1 by Sw, and cyclic shifts s 7→ ˆs by Cy.

Example (ﬂattening algorithm)

Cy Cy Cy 2 1) 264 −→ 642 −→Sw 552 −→ 525 −→Sw 345 −→ 534 −→Sw 444. (This implies ...what?) Cy 2 Cy 2 2) 514 −→Sw 244 −→ 424 −→Sw 334 −→ 433. (Similarly, this implies...?)

This proves the Average Theorem! (why??)

Yuki Takahashi (UC Irvine) November 12, 2015 29 / 51 Test Vector

Deﬁnition (Test Vector) p−1 Let s = {ak }k=0 be a nonnegative sequence. Then we call the vector

(0 + a0, 1 + a1, ··· , (p − 1) + ap−1) mod p

the test vector of s.

Example Let s = 6424. Then (0 + 6, 1 + 4, 2 + 2, 3 + 4) = (6, 5, 4, 7), so the test vector is (2, 1, 0, 3). Let s = 543. Then (0 + 5, 1 + 4, 2 + 3) = (5, 5, 5), so the test vector is (2, 2, 2).

Yuki Takahashi (UC Irvine) November 12, 2015 30 / 51 Permutation Test

Theorem (The Permutation Test) p−1 Let s = {ak }k=0 be a nonnegative sequence, and let φs be the test vector of s. Then, s is a juggling sequence if and only if φs is a permutation.

Example Let s = 6424. Then the test vector is (2, 1, 0, 3), so 6424 is a juggling sequence. Let s = 444. Then the test vector is (1, 2, 0), so 444 is a juggling sequence. Let s = 433. Then the test vector is (1, 1, 2), so 433 is NOT a juggling sequence.

Yuki Takahashi (UC Irvine) November 12, 2015 31 / 51 Idea of the Proof

The proof is based on the ﬂattening algorithm.

The key fact is that: ”permutationability” is preserved under swapping and cyclic shift!!!

Example

Let s = 7531. Then s1,2 = 7441. The test vector of 7531 is (3, 2, 1, 0), and the test vector of 7441 is (3, 1, 2, 0).

Let s = 6534. Then s2,3 = 6552. The test vector of 6534 is (2, 2, 1, 3), and the test vector of 6552 is (2, 1, 3, 1).

Yuki Takahashi (UC Irvine) November 12, 2015 32 / 51 Proof of the Permutation test

WTS: s is a juggling sequence if and only if the test vector of s is a permutation.

Example (Flattening Algorithm and Test Vector)

Cy 1) 642 (0, 2, 1) −→Sw 552 (2, 0, 1) −→ 525 (2, 0, 1) Cy 2 −→Sw 345 (0, 2, 1) −→ 534 (2, 1, 0) −→Sw 444 (1, 2, 0). (This implies ...what?) Cy 2 2) 514 (2, 2, 0) −→Sw 244 (2, 2, 0) −→ 424 (1, 0, 0) Cy 2 −→Sw 334 (0, 1, 0) −→ 433 (1, 1, 2). (Similarly, this implies...?)

This proves the Permutation Test! :) (why??)

Yuki Takahashi (UC Irvine) November 12, 2015 33 / 51 Corollary

Corollary (Vertical Shifts) p−1 Let s = {ak }k=0 be a sequence of nonnegative integers. Let d be an 0 p−1 integer such that s = {ak + d}k=0 is a sequence of nonnegative integers. Then s is a juggling sequence if and only if s0 is a juggling sequence. We call this operation the vertical shifts.

Example Let s = 441. Then s0 = 996 is also a juggling sequence.

Yuki Takahashi (UC Irvine) November 12, 2015 34 / 51 Converse of the Average Theorem

Theorem Given a ﬁnite sequence of nonnegative integers whose average is an integer, there is a permutation of this sequence that is a juggling sequence.

Example 43210 is NOT a juggling sequence, but 01234, 02413, 03142 are all juggling sequences.

Yuki Takahashi (UC Irvine) November 12, 2015 35 / 51 Scramblable Juggling Sequences

Deﬁnition (Scramblable sequences) Scramblable sequences are juggling sequences that stay juggling sequence no matter how you rearrange their elements. (Example: 3333, 1999, 147 are all scramblable sequences)

Theorem A ﬁnite sequence of nonnegative integers is a scramblable juggling p−1 sequence of period p if and only if it is of the form {ak p + c}k=0, where c and ak are nonnegative integers.

Yuki Takahashi (UC Irvine) November 12, 2015 36 / 51 Magic Juggling Sequences

Deﬁnition (Magic juggling sequence) A magic juggling sequence is a juggling sequence of some period p that contains every integer from 0 to p − 1 exactly once. (Example: 0123456 is a magic juggling sequence)

Theorem Let p and q be two positive integers such that p is odd, q > 1, and p is relatively prime to both q and q − 1. Then

p−1 {(q − 1)k mod p}k=0 is a magic juggling sequence.

Let q = 2. Then we see that 0123 ··· (p − 1) is a magic juggling sequence for any odd number p.

Yuki Takahashi (UC Irvine) November 12, 2015 37 / 51 How Many Ways to Juggle?

Theorem The number of all minimal b-ball juggling sequences of period p is

1 X p µ (b + 1)d − bd , p d d|p

where µ is the M¨obiusfunction.  1 if n has an even number of distinct prime factors,  µ(n) = −1 if n has an odd number of distinct prime factors, 0 if n has repeated prime factors.

Example: if b = 3, and p = 3, then there are 12 ways.

Yuki Takahashi (UC Irvine) November 12, 2015 38 / 51 Points of Intersection

Given a juggling sequence s, let cross(s) be the number of points of intersection of arcs in the juggling diagram.

Example (1)

Figure : s = 4. cross(s) = ...?

Yuki Takahashi (UC Irvine) November 12, 2015 39 / 51 Points of Intersection

Example (1)

Figure : s = 4. Then cross(s) = 3.

Yuki Takahashi (UC Irvine) November 12, 2015 40 / 51 Points of Intersection

Example (2)

Figure : s = 441. cross(s) = ...?

Yuki Takahashi (UC Irvine) November 12, 2015 41 / 51 Points of Intersection

Example (2)

Figure : s = 441. Then cross(s) = 4.

Yuki Takahashi (UC Irvine) November 12, 2015 42 / 51 Aﬃne Weyl Group A˜p−1

Deﬁnition p−1 Let s = {ai }i=0 be a juggling sequence. Deﬁne a map ϕs : Z → Z by

ϕs : i 7→ ai mod p + i − b,

where b is the number of balls juggled.

Example Let s = 552. Then b = 4, and we have

· · · −1 0 1 2 ··· · · · −1 0 1 2 ··· = ··· ϕ(−1) ϕ(0) ϕ(1) ϕ(2) ··· · · · −3 1 2 0 ···

For any juggling sequence s, ϕs is in fact a permutation of Z. Denote the set of all permutations arising in this way by A˜p−1.

Yuki Takahashi (UC Irvine) November 12, 2015 43 / 51 Aﬃne Weyl Group A˜p−1

Let us deﬁne simple reﬂection tk : Z → Z by  i + 1 for i = k mod p,  i 7→ i − 1 for p = k + 1 mod p, i o.w.

Given σ ∈ A˜p−1, let length(σ) be the smallest integer such that σ can be written as the product of this number of simple reﬂections.

Yuki Takahashi (UC Irvine) November 12, 2015 44 / 51 Aﬃne Weyl Group A˜p−1

Theorem Let s be a b-ball juggling sequence of period p. Then

cross(s) = (b − 1)p − length(ϕs ).

Example Let s = 441. Then b = 3, and

··· 0 1 2 ··· ϕ = . s ··· 1 2 0 ···

Then length(ϕs ) = 2. Therefore, cross(s) = (3 − 1) · 3 − 2 = 4.

Yuki Takahashi (UC Irvine) November 12, 2015 45 / 51 Multiplex Juggling

Multiplex juggling is a natural generalizations of the simple juggling patterns.

Deﬁnition (Multiplex Juggling Patterns) Multiplex juggling patterns satisfy the balls are juggled in a constant beat, that is, the throws occur at discrete equally spaced moments in time, patterns are periodic, and all the balls that get caught on a beat also get tossed on the same beat.

Yuki Takahashi (UC Irvine) November 12, 2015 46 / 51 Multiplex Juggling

Example

Figure : Juggling diagram of the multiplex juggling sequence [14]1.

Yuki Takahashi (UC Irvine) November 12, 2015 47 / 51 The Average Theorem for Multiplex Juggling

Theorem (The Average Theorem) The number of balls necessary to juggle a multiplex juggling sequence equals its ”average”.

Example (1+4)+1 The number of balls necessary to juggle [14]4 is 2 = 3.

Yuki Takahashi (UC Irvine) November 12, 2015 48 / 51 JuggleBuddies

Figure : 2015, October, Aldrich park, UCI Illuminations.

We are recruiting new members. Everyone is welcome! =]

Yuki Takahashi (UC Irvine) November 12, 2015 49 / 51 JuggleBuddies Information

Webpage: www.jugglebuddies.webs.com Facebook: https://www.facebook.com/groups/858609887507070/

Yuki Takahashi (UC Irvine) November 12, 2015 50 / 51 Thank you! :)

Yuki Takahashi (UC Irvine) November 12, 2015 51 / 51