Some Involutory Pascal Matrices Can Make More Involutory Pascal Matrices

Summary

There is an intuitive way to flip signs within a lower triangular matrix $n\times n$ matrix whose entries form the first n – 1 rows of Pascal’s triangle to cause the matrix to be its own multiplicative inverse. For any given positive integer n, we find how many unintuitive choices of flipped signs will also cause the n × n matrix to be its own multiplicative inverse by first seeing how such matrices may or may not fit into larger matrices from the same family.

Acknowledgment

We would like to thank the anonymous referees of both this iteration and a previous iteration of this paper for their very helpful suggestions, especially with respect to notation.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Keith Copenhaver

Keith Copenhaver ([email protected]) is an Assistant Professor of Mathematics at Eckerd College. He received his Ph.D. under Miklós Bóna from the University of Florida in 2019. He enjoys spending time with his family, doing combinatorics, video games, and ultimate frisbee.

Josh Hiller

Josh Hiller ([email protected]) is a discrete mathematician and Associate Professor of Mathematics and Computer Science at Adelphi University. He received his Ph.D. in Mathematics under James Keesling at the University of Florida in 2017. In his spare time, he can be found playing with his children or strolling through Central Park.

Andrew Velasquez-Berroteran

Andrew Velasquez-Berroteran ([email protected]) is currently a Ph.D. student at Binghamton University studying mathematics, planning to do research in group theory. For fun, he likes listening to alternative and rock music, and doing Sudoko and other logic puzzles.