Non-Associativity of the
Double Minus Operation
MADISON MICKEY
UNIVERSITY OF NEBRASKA AT KEARNEY
PRESENTED: 3/14/2017
JOINT WORK WITH J. HUANG AND J. XU
Double Minus Operation

Define binary operation: 𝑎 ө b = −a − b

Let 𝐶ө,𝑛,𝑟 be the number of distinct results with exactly r plus
signs from
𝑥0 ө 𝑥1 ө … ө 𝑥𝑛

Let 𝐶ө,𝑛 = 0 ≤𝑟 ≤𝑛+1 𝐶ө,𝑛,𝑟 for 𝑛 ≥ 0 which is the total number of
distinct results from the previous expression.

𝐶ө,𝑛 measures the non-associativity of the operation ө
Example (n = 3)
− − −𝑥0 − 𝑥1 − 𝑥2 − 𝑥3 = −𝑥0 − 𝑥1 + 𝑥2 − 𝑥3
− −𝑥0 − 𝑥1 − −𝑥2 − 𝑥3 = 𝑥0 + 𝑥1 + 𝑥2 + 𝑥3
− −𝑥0 − −𝑥1 − 𝑥2
− 𝑥3 = 𝑥0 − 𝑥1 − 𝑥2 − 𝑥3
−𝑥0 − − −𝑥1 − 𝑥2 − 𝑥3 = −𝑥0 − 𝑥1 − 𝑥2 + 𝑥3
−𝑥0 − −𝑥1 − −𝑥2 − 𝑥3
= −𝑥0 + 𝑥1 − 𝑥2 − 𝑥3
Cө,3,0 = 0
Cө,3,1 = 4 =
4
1
Cө,3,2 = 0
Cө,3,3 = 0
Cө,3,4 = 1 =
Cө,3
4
4
4
4
=
+
=5
1
4
--+++++
+-----+
-+--
Theorem (Huang, M., and Xu)
For 𝑛 ≥ 1 and 0 ≤ 𝑟 ≤ 𝑛 + 1:
𝐶ө,𝑛,𝑟 =
𝑛+1
𝑟
𝑛+1
−1
𝑟
0
𝑖𝑓 𝑛 + 𝑟 ≡ 1 𝑚𝑜𝑑 3 𝑎𝑛𝑑 𝑛 ≠ 2𝑟 − 2;
𝑖𝑓 𝑛 + 𝑟 ≡ 1 𝑚𝑜𝑑 3 𝑎𝑛𝑑 𝑛 = 2𝑟 − 2;
𝑖𝑓 𝑛 + 𝑟 ≢ 1 𝑚𝑜𝑑 3 .
Theorem (Huang, M., and Xu)
For 𝑛 ≥ 1 we have
2𝑛+1 −1
𝐶ө,𝑛 =
3
2𝑛+1 −2
3
,
𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
,
𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛.
Proof by Example


6
0
+
6
6
+
6
1
=?
+
6
4
6
2
=?
6
5
=?
𝑥5 +
6
6
+
Using the binomial expansion:
6
0
6
1
𝑥1 +
6
2
𝑥2 +
6
2
6
3
+

(1 + 𝑥)6 =

𝑥 = 1:

𝑥 = −1:

Add together and divide by 2:


6
3
6
0
6
1
+
6
0
+
−
+
6
1
+
+
6
2
−
6
3
6
4
+
6
3
6
4
𝑥3 +
+
6
4
6
5
−
6
0
+
+
6
5
𝑥4 +
𝑥6
6
6
+
6
2
6
5
+
6
6
6
4
+
6
6
Result based on ±1 = 1 and 1 + (−1) = 0 (Second Roots of Unity)
Used Third Roots of Unity to calculate binomial coefficient totals

13 = 1, 𝜔3 = 1, (𝜔2 )3 = 1 and 1 + 𝜔 + 𝜔2 = 0
Examining Non-Associativity

Related to Sequence A000975 in the OEIS (On-line Encyclopedia of Integer Sequences)

This sequence is characterized by a recurrence relation.

This sequence also gives binary operations with alternating ones and zeros.

A(1)
=
1
=
12
=
1 × 20
A(2)
=
2
=
102
=
1 × 21 + 0 × 20
A(3)
=
5
=
1012
=
1 × 22 + 0 × 21 + 1 × 20
A(4)
=
10 =
10102
=
1 × 23 + 0 × 22 + 1 × 21 + 0 × 20
A(5)
=
21 =
101012 =
1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20
This sequence occurs in other interesting places.
THANK YOU!