Labeled partitions with colored permutations

Chen, William Y. C.; Gao, Henry Y.; He, Jia

doi:10.1016/j.disc.2009.06.006

Cited by 4 publications

(3 citation statements)

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“…However, to our knowledge, there are no published Gray codes for coloured permutations. This is surprising as the combinatorial [4,5,7,17,18] and algebraic [2,3,22] properties of coloured permutations have been of considerable interest. Work on the latter is due to the group theoretic interpretation of P k (n) as the wreath product of the cyclic and symmetric group, Z k S n .…”

Section: Combinatorial Generationmentioning

confidence: 99%

“…If we iterate over the permutations using a prefix-reversal Gray code, then successive Hamilton paths differ in a single edge. For example, the edges in 12345678 and 43215678 are identical, except that the former includes (4, 5) while the latter includes (1,5). Thus, the cost of each Hamilton cycle can be updated from permutation to permutation using one addition and subtraction.…”

Section: Combinatorial Generationmentioning

confidence: 99%

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A Hamilton Cycle in the $k$-Sided Pancake Network

Cameron

Sawada

Williams

2021

Preprint

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We present a Hamilton cycle in the k-sided pancake network and four combinatorial algorithms to traverse the cycle. The network's vertices are coloured permutations π = p1p2 • • • pn, where each pi has an associated colour in {0, 1, . . . , k−1}. There is a directed edge (π1, π2) if π2 can be obtained from π1 by a "flip" of length j, which reverses the first j elements and increments their colour modulo k. Our particular cycle is created using a greedy min-flip strategy, and the average flip length of the edges we use is bounded by a constant. By reinterpreting the order recursively, we can generate successive coloured permutations in O(1)-amortized time, or each successive flip by a loop-free algorithm. We also show how to compute the successor of any coloured permutation in O(n)-time. Our greedy min-flip construction generalizes known Hamilton cycles for the pancake network (where k = 1) and the burnt pancake network (where k = 2). Interestingly, a greedy max-flip strategy works on the pancake and burnt pancake networks, but it does not work on the k-sided network when k > 2.

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Section: Combinatorial Generationmentioning

confidence: 99%

Section: Combinatorial Generationmentioning

confidence: 99%

A Hamilton Cycle in the $k$-Sided Pancake Network

Cameron

Sawada

Williams

2021

Preprint

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“…2]. For generalizations to other groups, see [3,7,13,10,9,12]. In this paper we study signed major index enumerators and other related polynomials for arc permutations, both in the symmetric group S n and in the hyperoctahedral group B n .…”

Section: Introductionmentioning

confidence: 99%

Signed arc permutations

Elizalde

Roichman

2015

Journal of Combinatorics

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Arc permutations, which were originally introduced in the study of triangulations and characters, have recently been shown to have interesting combinatorial properties. The first part of this paper continues their study by providing signed enumeration formulas with respect to their descent set and major index. Next, we generalize the notion of arc permutations to the hyperoctahedral group in two different directions. We show that these extensions to type B carry interesting analogues of the properties of type A arc permutations, such as characterizations by pattern avoidance, and elegant unsigned and signed enumeration formulas with respect to the flag-major index.

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