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Factorization of banded permutations. (English) Zbl 1270.05003

Summary: 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.

MSC:

05A05 Permutations, words, matrices
20B99 Permutation groups
15A23 Factorization of matrices
15B99 Special matrices
65T50 Numerical methods for discrete and fast Fourier transforms
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References:

[1] Chase Albert, Chi-Kwong Li, Gilbert Strang, and Gexin Yu, Permutations as product of parallel transpositions, SIAM J. Discrete Math. 25 (2011), no. 3, 1412 – 1417. · Zbl 1237.05002 · doi:10.1137/100807478
[2] Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. · Zbl 1110.05001
[3] Martianus Frederic Ezerman and Michael Daniel Samson, Factoring Permutation Matrices into a Product of Tridiagonal Matrices (2010), available at arXiv:1007.3467.
[4] Jacob E. Goodman, Proof of a conjecture of Burr, Grünbaum, and Sloane, Discrete Math. 32 (1980), no. 1, 27 – 35. · Zbl 0444.05029 · doi:10.1016/0012-365X(80)90096-5
[5] Gilbert Strang, Fast transforms: banded matrices with banded inverses, Proc. Natl. Acad. Sci. USA 107 (2010), no. 28, 12413 – 12416. · Zbl 1205.65348 · doi:10.1073/pnas.1005493107
[6] Gilbert Strang, Groups of banded matrices with banded inverses, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4255 – 4264. · Zbl 1241.15011
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