Minimal factorizations of permutations into star transpositions

Abstract

We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form (1 i). This generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored. Our result exhibits an unexpected and simple symmetry of star factorizations that has yet to be explained in a satisfactory manner.

Distance formula and shortest paths for the (n,k)-star graphs

“…By copy, we mean copy the exchange of the symbols. See Table 1 for an example when the chosen associated permutation of [4,2,11,1,5,12,6,7,14,3] is [4,2,11,1,5,12,6,7,14,3,8,9,13,10,15]. Clearly q is an associated permutation of ½1; 2; .…”

“…; k and we are done. (Table 2 provides another example when we pick [4,2,11,1,5,12,6,7,14,3,15,10,9,13,8] to be the associated permutation.) h It follows immediately from the above proposition that finding distances in S n;k reduces to finding distances in S n .…”

“…It is easy to see that such an extended permutation is indeed a permutation and that each of the external symbols is in a distinct cycle in the cycle decomposition of p E . In the above example, note that we purposely write each of 11,12,13,14,15 in the last position of the cycle containing it. Indeed for the cycle containing each external symbol, we will write the external symbol last in the cycle.…”

“…We first note that given permutations p and q, the distance between them is the same as the distance between a permutation a and the identity, as S n is vertex-transitive, where a is the permutation of the symbols in p with respect to q. [4,2,11,1,5,12,6,7,14,3,8,10,9,13,15] and q to be [1,2,3,4,5,6,7,8,9,10,11,12,13,15,14]. Then a, the corresponding permutation of the symbols in p with respect to q, is precisely 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 4 2 11 1 5 12 6 7 14 3 8 10 9 13 15 :…”

“…In this paper, we include the fixed points in the decomposition. So (2), (5), (13) and (15) are included. Moreover, we refer to these cycles of length one as trivial cycles.…”