Factorization of banded permutations

TL;DR

This paper proves that any banded permutation of bandwidth w can be factored into at most 2w-1 permutations of bandwidth 1, confirming a conjecture by Gilbert Strang.

Contribution

It establishes an upper bound on the minimal number of bandwidth 1 permutations needed to factorize a banded permutation, extending to infinite and cyclic cases.

Findings

Proved the conjecture of Gilbert Strang.

Established an upper bound of 2w-1 for factorization.

Extended results to infinite and cyclically banded permutations.

Abstract

We consider the factorization of permutations into bandwidth 1 permutations, which are products of mutually nonadjacent simple transpositions. We exhibit an upper bound on the minimal number of such factors and thus prove a conjecture of Gilbert Strang: a banded permutation of bandwidth $w$ can be represented as the product of at most $2w-1$ permutations of bandwidth 1. An analogous result holds also for infinite and cyclically banded permutations.