Robert Parviainen
We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.
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In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.
In this paper we provide a unified combinatorial approach to establish a connection between Stirling permutations, cycle structures of permutations and perfect matchings. The main tool of our investigations is MY-sequences. In particular, we discover that the Eulerian polynomials have a simple combinatorial interpretation in terms of some statistics on MY-sequences.
A universal cycle for permutations of length $n$ is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length $n$, and containing all permutations of length $n$ as factors. It is well known that universal cycles for permutations of length $n$ exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, …
Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating method is proposed in this paper. Accurate enumerating formulae are provided when $k=2,3$.
We prove the Complete nontrivial cycle-intersection theorem for systems of permutations.
In this paper, we study the occurrence of patterns in the cycle structures of permutations.
We give a generating function for the number of unimodal permutations with a given cycle structure.
A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the Euler numbers, and we study the distribution of some statistics on them, as well as on up-down permutations, on all permutations, and on a generalization of cycle-up-down permutations. The statistics include the number of cycles of even and odd length, the number of left-to-right minima, and the number of extreme elements.
We discuss both simple and more subtle connections between the numbers of permutations and full cycles with some restrictions,in particular, between the numbers of permutations and full cycles with prescribed up-down structure.
Let $k$ be a nonnegative integer, and let $α$ and $β$ be two permutations of $n$ symbols. We say that $α$ and $β$ $k$-commute if $H(αβ, βα)=k$, where $H$ denotes the Hamming metric between permutations. In this paper, we consider the problem of finding the permutations that $k$-commute with a given permutation. Our main result is a characterization of permutations that $k$-commute with a given permutation $β$ in terms of blocks in cycles in the decomposition of $β$ as a product of disjoint cycles. Using this characterization, we provide formulas for the number of permutations that $k$-commute …
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit formulas, recurrence relations, and ordinary and exponential generating functions. A generalization to more than one fixed length is also considered.
The density, denoted by $κ(n,r)$, of permutations having no cycles of length less than $r+1$ in a symmetric group $\mathrm{S}_n$ is explored. New asymptotic formulas for $κ(n,r)$ are obtained using the saddle-point method when $5\leq r< n$ and $n\to\infty$.
A remark on continued fractions for permutations and D-permutations with a weight $-1$ per cycle
该研究证明了每个循环带权重-1时，带多独立统计量的排列与D-排列的普通生成函数存在简洁连分数形式，并借助关联模2循环数与不动点、循环峰谷、交叉数的引理完成证明。
On a family of universal cycles for multi-dimensional permutations
本文提出一种基于贪心算法构造多维排列通用循环的方法，推广了 Gao 等人的排列 u-cycle 构造，并指出 d 维矩阵也存在 u-cycle。
A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham.
We prove bijectively that the total number of cycles of all even permutations of $[n]=\{1,2,...,n\}$ and the total number of cycles of all odd permutations of $[n]$ differ by $(-1)^n(n-2)!$, which was stated as an open problem by Miklós Bóna. We also prove bijectively the following more general identity: $$\sum_{i=1}^n c(n,i)\cdot i \cdot (-k)^{i-1} = (-1)^k k! (n-k-1)!,$$ where $c(n,i)$ denotes the number of permutations of $[n]$ with $i$ cycles.
This paper initiates the study of shortening universal cycles (u-cycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for $n$-permutations exist of lengths $n!+(1-k)(n-1)$ for $k=0,1,\ldots,(n-2)!$.
Pattern-restricted cyclic permutations with a pattern-restricted cycle form
本文研究了避免单调递减排列的循环排列，在循环标准形式避免特定长度模式或所有循环旋转避免特定长度模式的情况下，分别通过连分数生成函数给出答案并枚举了相关情形。
Enumerating two permutation classes by the number of cycles
本文按循环数对两个置换类 Av_n(312,4321) 和 Av_n(321,4123) 进行枚举，并针对若干统计量细化结果。
On the cycle structure of the symmetric tensor power of permutations
本文研究置换σ的循环分解与其对称张量幂σ^{⊙k}的循环分解之间的关系，给出了计算不动点数量的公式，并在σ至多含有一个长度大于1的循环时给出了s-循环数量的公式。
On Cycles in Multiset Permutations, Parking Functions, and Related Structures
本文研究多重集排列和停车函数中的环，通过代数和分析方法给出精确与渐近结果，并揭示终端闭包与循环点之间的意外对应关系。
In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of $n$ 3-cycles that avoid the pattern 231 (equivalently 312) is given by $3^{n-1}$, while the generating function for the number of those that avoid the pattern 132 (equivalently 213) is given by a formula involving the generating functions for the well-known Motzkin numbers and Catalan numbers. The number of permutations composed of $n$ 3-cycles that avoid the pattern 321 …
The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length $n$ and edges that are permutations of length $n+1$ in which an edge $a_1\cdots a_{n+1}$ would connect the standardization of $a_1\cdots a_n$ to the standardization of $a_2\cdots a_{n+1}$. We examine properties of this graph to determine where directed cycles can exist, to count the number of directed $2$-cycles within the graph, and to enumerate the vertices that are contained within closed walks and directed cycles of more general lengt…
The goal of this paper is to solve Problem 481 from the list of research problems in the special issue of Discrete Mathematics dedicated to the Banff International Research Station workshop on "Generalizations of de Bruijn Cycles and Gray Codes" in 2004. Overlap cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally in 2010 by Godbole et al. In this paper we prove that s-overlap cycles for k-permutations of [n] exist for all k<n.
Counting pairs of cycles whose product is a permutation with restricted cycle lengths
本文研究乘积循环长度受限时 $N$-循环对的计数，给出精确与渐近公式，并据此证明与恒等排列块转置距离至少为 $(n+1)/2$ 的排列数量具有较高下界。
Descent set distribution for permutations with cycles of only odd or only even lengths
本文证明对称群中仅含奇长度循环与仅含偶长度循环的排列数等式在下降集层面的细化，并给出奇数阶情形的变体，方法基于特征标生成函数与高阶 Lie 特征标新恒等式。
We prove that the Mahonian-Stirling pairs of permutation statistics $(\sor, \cyc)$ and $(\inv, \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sor…
Cyclic permutations avoiding patterns in both one-line and cycle forms
本文研究长度为3的模式对(σ,τ)，枚举对称群中在单行记号避免σ且在循环记号避免τ的循环排列数量。
Answering a question of Bona, it is shown that for n>1 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1,2,...,n} is 1/2 if n is odd and 1/2 - 2/(n-1){n+2) if n is even. Another result concerns the generating function P_h(q) for the number of cycles of the product (1,2,...,n)w, where w ranges over all permutations of 1,2,...,n of cycle type h. A formula is obtained for P_h(q) from which it is proved that the zeros of P_h(q) have real part 0.
Let $γ_n$ be the permutation on $n$ symbols defined by $γ_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations $β$ of $n$ in which the numbers from 1 to $n$ are colored with $p$ colors such that two elements in a same cycle have the same color. We show that the proportion of colored permutations such that $γ_n β^{-1}$ is a long cycle is given by the very simple ratio $\frac{1}{n- p+1}$. Our proof is bijective and uses combinatorial objects such as partitioned hypermaps and thorn trees. This formula is actually equivalent to the proportio…
We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of the random permutation of $\{1,2,...,n\}$ obtained as product of two uniformly random $n$-cycles.
Continued fractions for cycle-alternating permutations
该研究针对循环交替排列的计数多元多项式得到Stieltjes型连分数，引入其推广结构交替Laguerre有向图并给出对应计数多项式的指数生成函数，还从图的角度解释了相关连分数对应的两类Rogers矩阵。
Shortened universal cycles for permutations
本文通过构造性证明验证了 Kitaev 等人关于利用不可比较元素将排列通用循环缩短至 $n!-i(n-1)$ 的猜想，并提出一种新的排列通用循环构造方法。
The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation $π= π_{1} π_{2} ... π_{n+1}$ there is a directed edge from the standardization of $π_{1} π_{2} ... π_{n}$ to the standardization of $π_{2} π_{3} ... π_{n+1}$. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.
In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. This algorithm is very similar to the standard method for generating a random permutation, but is less well known. We consider both methods in a unified way, and discuss their relation with exhaustive generation methods. We analyse several random variables associated with the algorithms and find their grand probability generating functions, which gives easy access to moments and limit laws.
Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and E-permutations) for (median) Genocchi numbers. They further conjecture that the distributions of cycle numbers over the two models are equal. In a follow-up, Eu et al. (arXiv:2103.09130) further proved the gamma-positivity of the descent polynomials of even-odd descent permutations, which are in bijection with E-permutations by Foata's fundamental transformation. This …
Enriched Cycle Structures and Roots of Permutations
本文通过定义富集置换，揭示了 $r$-正则置换与富集 $r$-循环置换之间的双射关系，从而证明了 $r$-th 根概率的单调性质，并回答了 Bóna-McLennan-White 的问题。
Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations. Hoffman established a link between Springer numbers, snakes, and some polynomials related with the successive derivatives of trigonometric functions. The goal of this article is to give further combinatorial properties of derivative polynomials, in terms of snakes and other objects: cycle-alternating permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We obtain the generating functions,…
We study joint distributions of cycles and patterns in permutations written in standard cycle form. We explore both classical and generalised patterns of length 2 and 3. Many extensions of classical theory are achieved; bivariate generating functions for inversions, ascents, descents, 123s, valleys, 1'-2-1s; closed forms forms for avoidance of peaks, 2-3-1s, 1-2-3s, 2'-1-2s and 1'-2-1s; bijective proofs of Wilf-equivalences. We also derive some results about standard pattern occurrence, such as continued fractions for the generating functions for occurrences of valleys and the pattern 123. The…
In this paper, we enumerate the pairs of permutations that are long cycles and whose product has a given cycle-type. Our main result is a simple relation concerning the desired numbers for a few related cycle-types. The relation refines a formula of the number of pairs of long cycles whose product has $k$ cycles independently obtained by Zagier and Stanley relying on group characters, and was previously obtained by Féray and Vassilieva by counting some colored permutations first and then relying on some algebraic computations in the ring of symmetric functions. Our approach here is simpler and…
Cycle structures and parities of linearized permutation polynomials
本文刻画线性化置换多项式的循环结构与奇偶性，为Mullen和Vaughan1988年的问题提供迪克森矩阵形式表述，推导奇偶性判据并给出判定算法，应用于编码、密码等领域。
We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc. with certain minimality and transitivity conditions imposed on the factors. The method is to encode such factorizations as planar maps with certain "descent structure" and apply a new combinatorial decomposition to make their enumeration more manageable. We apply our technique to count factorizations of permutations with one or two parts, extending earlier w…
We characterize the class of cycle decompositions that can achieve all Young tableau shapes (except the trivial ones with a single row or a single column) under the Robinson--Schensted--Knuth (RSK) correspondence, a property that we call RSK-completeness. We prove that for even $n$, cyclic permutations comprise the only fixed cycle decomposition that is RSK-complete. For odd $n$, cyclic permutations and almost cyclic permutations which have a cycle of length $n-1$ are the only RSK-complete cycle decompositions.
We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables.
In this paper we adapt techniques used by Ahlswede and Khachatrian in their proof of the Complete Erdős-Ko-Rado Theorem to show that if $n \geq 2t+1$, then any pairwise $t$-cycle-intersecting family of permutations has cardinality less than or equal to $(n-t)!$. Furthermore, the only families attaining this size are the stabilizers of $t$ points, that is, families consisting of all permutations having $t$ 1-cycles in common. This is a strengthening of a previous result of Ku and Renshaw and supports a recent conjecture by Ellis, Friedgut and Pilpel concerning the corresponding bound for $t$-in…
Generation of Cycle Permutation Graphs and Permutation Snarks
本文提出算法生成所有非同构的循环置换图，包括非哈密顿的循环置换图和置换snarks，提高了之前结果的上限，并提供了关于置换snarks的下界和猜想的证据。
The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $Ω$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples. In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is the orbit…
Balanced Gray Codes for Permutations and Rainbow Cycles for Associahedra
本研究解决了排列的平衡转置格雷码构造问题，并推广了彩虹环概念，为结合多面体和排列多面体提供了新的彩虹环构造，同时补充了偶数情况下的平衡格雷码。
Any permutation polynomial is an $ n $-cycle permutation. When $n$ is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for $ n $-cycle permutations, which mainly are of the form $ x^rh(x^s) $. We then propose unified constructing methods including recursive ways and a cyclotomic way for $ n $-cycle permutations of such form. We demonstrate our approaches by c…
Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= Σ_τλ_f(τ)τ$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(π,R)$ is an occurrence of the permutation pattern $π$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(π,R)$, we have $λ_p(τ) = (-1)^{|τ|-|π|}p^{\star}(τ)$ …
Let $β$ be any permutation on $n$ symbols and let $c(k, β)$ be the number of permutations that $k$-commute with $β$. The cycle type of a permutation $β$ is a vector $(c_1, \dots, c_n)$ such that $β$ has exactly $c_i$ cycles of length $i$ in its disjoint cycle factorization. In this article we obtain formulas for $c(k, β)$, for some cycle types. We also express these formulas in terms of integer sequences as given in "The On-line Encyclopedia of Integer Sequences" (OEIS). For some of these sequences we obtain either new interpretations or relationships with sequences in the OEIS database.
On a conjecture about pattern avoidance of cycle permutations
本文研究轮换排列的模式避免问题，证明了对称群中单行形式避免2431且所有轮换形式避免1324的轮换排列数量为第(n-1)个Pell数，肯定回答了Archer等人的猜想。
In this paper, we first present combinatorial proofs of a kind of expansions of the Eulerian polynomials of types A and B, and then we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles.
We describe a new method for finding patterns in permutations that produce a given pattern after the permutation has been passed once through a stack. We use this method to describe West-3-stack-sortable permutations, that is, permutations that are sorted by three passes through a stack. We also show how the method can be applied to the bubble-sort operator. The method requires the use of mesh patterns introduced by Brändén and Claesson (2011), as well as a new type of generalized pattern we call a decorated pattern.
Let $G$ be a graph on $n$ vertices, labeled $v_1,\ldots,v_n$ and $π$ be a permutation on $[n]:=\{1,2,\cdots, n\}$. Suppose that each pebble $p_i$ is placed at vertex $v_{π(i)}$ and has destination $v_i$. During each step, a disjoint set of edges is selected and the pebbles on each edge are swapped. 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 prove that $rt(C_n, π)\le n-1$ for any permutation on $n$-cycle $C_n$ and conjecture that for $n \geq 5$, if $rt(C_n, π) = n-1$, then $π= (123\cdots n)$ …
Pattern-restricted permutations of small order
本文利用Catalan与Motzkin生成函数枚举小阶数的132-避免排列，回答了Bóna和Smith的公开问题，并刻画了仅含3-循环、2-循环和不动点的231-避免排列。
Previous work has studied the pattern count on singly restricted permutations. In this work, we focus on patterns of length 3 in multiply restricted permutations, especially for double and triple pattern-avoiding permutations. We derive explicit formulae or generating functions for various occurrences of length 3 patterns on multiply restricted permutations, as well as some combinatorial interpretations for non-trivial pattern relationships.
Construction of Permutation Polynomials of Certain Specific Cycle Structure over Finite Fields
本文针对奇数阶有限域，构造了具有特定圈结构的置换二项式和置换三项式族，包括具有一个不动点且其余元素构成等长互斥圈的置换，以及具有特定频率的多长度圈置换多项式。
In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and combinatorial identities for the counting sequence, mixed Stirling numbers of the first kind. In this comprehensive study we consider further the conditions on the length of the cycles, $r$-mixed Stirling numbers and the connection to Bell polynomials.
On cycles in monotone grid classes of permutations
本文研究行列图无多环的单调网格类排列结构，提出M-和分解，刻画良拟序条件，证明多数网格类有限基并反驳2006年猜想。
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