Permutations as Product of Parallel Transpositions

Albert, Chase; Li, Chi-Kwong; Strang, Gilbert; Yu, Gexin

doi:10.1137/100807478

Cited by 6 publications

(11 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A beautiful proof is given by Greta Panova [6], using the "wiring diagram" to decide the sequence of exchanges in advance. A second proof [1] by Albert, Li, Strang, and Yu confirms that the algorithm illustrated above also has N ≤ 2w − 1. A third proof is proposed by Samson and Ezerman [9].…”

Section: Banded Permutation Matricesmentioning

confidence: 63%

See 1 more Smart Citation

Groups of banded matrices with banded inverses

Strang

2011

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. A product A = F 1 . . . F N of invertible block-diagonal matrices will be banded with a banded inverse: A i j = 0 and also (A −1 ) ij = 0 for |i−j| > w. We establish this factorization with the number N controlled by the bandwidths w and not by the matrix size n. When A is an orthogonal matrix, or a permutation, or banded plus finite rank, the factors F i have w = 1 and we find generators of that corresponding group. In the case of infinite matrices, the A = LP U factorization is now established but conjectures remain open.

show abstract

Section: Banded Permutation Matricesmentioning

confidence: 63%

“…Greta Panova has found a beautiful proof [6]. Other constructions [1], [9] also yield N ≤ 2w − 1. (3) Banded plus finite rank.…”

Section: −1 Ijmentioning

confidence: 96%

Groups of banded matrices with banded inverses

Strang

2011

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this paper we will prove this conjecture. This conjecture has been proven independently later also by Albert,Li,Strang and Yu in [1] and Ezerman and Samson in [3].…”

Section: Introductionmentioning

confidence: 62%

“…For any permutation π, let k be the minimal number for which π = π (1) · · · π (k) , where π (i) are permutations of bandwidth 1. Then there exists a reduced decomposition of π = s (1) i1 s (1) i2 · · · s (k)…”

Section: Hook Wiring Diagramsmentioning

confidence: 99%

Factorization of banded permutations

Panova

2012

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

“…Lemma 14. If a permutation π has a pebble p 0 with win(p 0 ) = (p 0 , X, W ) and π = ({z}, p 0 , X, W ), then one of the following must be true (note that X could be empty) (1) if s(z) = c > 0, then X is an isolated block and W can be partitioned into isolated blocks and blocks with heads, say W 1 , W 2 , . .…”

Section: 2mentioning

confidence: 99%

Extremal Permutations in Routing Cycles

He¹,

Valentin²,

Yin³

et al. 2016

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

Let G be a graph whose vertices are labeled 1, . . . , n, and π be a permutation on [n] := {1, 2, · · · , n}. A pebble p i that is initially placed at the vertex i has destination π(i) for each i ∈ [n]. At each step, we choose a matching and swap the two pebbles on each of the edges. Let rt(G, π), the routing number for π, be the minimum number of steps necessary for the pebbles to reach their destinations.Li, Lu, and Yang proved that rt(C n , π) ≤ n − 1 for every permutation π on the n-cycle C n and conjectured that for n ≥ 5, if rt(C n , π) = n − 1, then π = 23 · · · n1 or its inverse. By a computer search, they showed that the conjecture holds for n < 8. We prove in this paper that the conjecture holds for all even n ≥ 6.

show abstract

Permutations as Product of Parallel Transpositions

Cited by 6 publications

References 3 publications

Groups of banded matrices with banded inverses

Groups of banded matrices with banded inverses

Factorization of banded permutations

Extremal Permutations in Routing Cycles

Contact Info

Product

Resources

About