A proof of the Loehr-Warrington amazing TEN to the power n conjecture

With Shalosh B. Ekhad and Doron Zeilberger.

Also available at arXiv:math.CO/0509347.

We prove, via 30 seconds of Maple computation, that there are 10ⁿ words in the alphabet {3,-2} of length 5n, sum 0, and such that every factor that sums to 0 and that starts with a 3 may not be immediately followed by a -2.

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Related links:

The Maple package TEN, which proved the theorem.
Input and output for TEN.
About the 2003 college football rankings fiasco, Will Poole (then a cornerback for snubbed USC) said “I learned a little about life in general. You can't let a computer make decisions for you. Computers are going to take over. The next thing you know, everybody is going to be out of a job. Computers are going to play football. If you let computers run the world, what are humans going to do?” Don't worry Will, there are now two computer-free proofs for (generalizations of) the 10ⁿ Theorem:
- “A human proof for a generalization of Shalosh B. Ekhad's 10ⁿ lattice paths theorem,” by Nick Loehr, Bruce Sagan, and Greg Warrington; and
- “Cylindrical lattice paths and the Loehr-Warrington ten to the power n conjecture,” by Jonas Sjöstrand, European Journal of Combinatorics 28 (2007), 774–780.
- See also Doron's rendition of Jonas' proof in “Another proof that Euler missed: Jonas Sjöstrand's amazingly simple (and lovely!) proof of the no-longer-so-amazing Loehr-Warrington lattice paths conjecture.”