A versatile combinatorial approach of studying products of long cycles in symmetric groups

Chen, Ricky X. F.

doi:10.1016/j.aam.2021.102283

Cited by 5 publications

(5 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Various proofs of combinatorial nature have been presented next. For instance, Cori, Marcus and Schaeffer [4], Chen and Reidys [7], and Chen [5]. In particular, a versatile combinatorial approach was presented in Chen [5] to deal with a variety of problems concering products of two long cycles.…”

Section: One-face Hypermapsmentioning

confidence: 99%

“…For instance, Cori, Marcus and Schaeffer [4], Chen and Reidys [7], and Chen [5]. In particular, a versatile combinatorial approach was presented in Chen [5] to deal with a variety of problems concering products of two long cycles. In Féray and Vassilieva [12], a refinement of the problem considered in Corollary 3.2 was studied, that is, enumerating the pairs of long cycles whose product has a given cycle-type.…”

Section: One-face Hypermapsmentioning

confidence: 99%

“…Combinatorial approaches attacking this topic have also received lots of interests. We refer to Bernardi [2], Chapuy [3], Goupil and Schaeffer [13], Goulden and Nica [14], Morales and Vassilieva [19], Chen and Reidys [7], Chen [5] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Explicit formulas for hypermaps and maps

Chen¹

2023

Preprint

View full text Add to dashboard Cite

The study of hypermaps and maps is ubiquitous, as they are closely connected with geometry, mathematical physics, free probability and algebra. A universal recursion generalizing a fundamental identity of Frobenius which enumerates factorizations of a permutation in group algebra theory has been discovered by the author recently. Here we apply the recursion to study hypermaps as well as maps and obtain a plethora of results in a unified way. For instance, we succinctly provide a short proof of the celebrated Harer-Zagier formula and present a general explicit formula for one-face hypermaps. As special cases of the latter, we obtain simple explicit formulas for the numbers of ways of expressing a long cycle as a product of a permutation of cycle-type [1 p , n − p] and, respectively, [p, n − p] and a permutation with m cycles for any p and m. To the best of our knowledge, only the cases for p = 0, 1 are known before.

show abstract

Section: One-face Hypermapsmentioning

confidence: 99%

Section: One-face Hypermapsmentioning

confidence: 99%

See 1 more Smart Citation

Explicit formulas for hypermaps and maps

Chen¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…We also employ Ψ = (s, f ) as a shorthand for the explicit two-row array representation if there is no confusion about which is the left-most element s 0 ∈ A. The idea of interpreting two-row arrays in such a way follows from plane permutations [5,6].…”

Section: Two-row Arrays: a Tool To Study Factorizationsmentioning

confidence: 99%

“…The latter states that the probability for two given labels to be contained in distinct cycles of the product of two random long cycles is 1/2 when n is odd, and has a surprising application in genome rearrangement problem concerning block-interchange distance [3]. Subsequent studies of tracking label distribution include, for instance, Bernardi, Du, Morales and Stanley [2], Bóna and Pittel [4], Chen [5], Féray and Rattan [10]. Here we add a new contribution to the subject.…”

Section: Introductionmentioning

confidence: 99%

On products of permutations with the most uncontaminated cycles by designated labels

Chen¹

2022

Preprint

View full text Add to dashboard Cite

There is a growing interest in studying the distribution of certain labels in products of permutations since the work of Stanley addressing a conjecture of Bóna. This paper is concerned with a problem in that direction. Let D be a permutation on the set [n] = {1, 2, . . . , n} and E ⊂ [n]. Suppose the maximum possible number of cycles uncontaminated by the E-labels in the product of D and a cyclic permutation on [n] is θ (depending on D and E). We prove that for arbitrary D and E with few exceptions, the number of cyclic permutations γ such that D • γ has exactly θ − 1 E-label free cycles is at least 1/2 that of γ for D • γ to have θ E-label free cycles, where 1/2 is best possible. An even more general result is also conjectured.

show abstract

On products of permutations with the most uncontaminated cycles by designated labels

Chen

2023

J Algebr Comb

View full text Add to dashboard Cite

A versatile combinatorial approach of studying products of long cycles in symmetric groups

Cited by 5 publications

References 26 publications

Explicit formulas for hypermaps and maps

Explicit formulas for hypermaps and maps

On products of permutations with the most uncontaminated cycles by designated labels

On products of permutations with the most uncontaminated cycles by designated labels

Contact Info

Product

Resources

About