A deck of $n$ cards are shuffled by repeatedly taking off the top card, flipping it with probability $1/2$, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group $B_n$ of signed permutations. We show that the eigenvalues of the transition probability matrix are $0,1/n,2/n,\ldots,(n-1)/n,1$ and the multiplicity of the eigenvalue $i/n$ is equal to the number of the {\em signed} permutation having exactly $i$ fixed points. We show the similar results also for the colored permutations. Further, we show that the mixing time of this Markov chain is $n\log n$, same as the ordinary 'top-to-random' shuffles without flipping the cards. The cut-off is also analyzed by using the asymptotic behavior of the Stirling numbers of the second kind.
The notion of shuffle-compatible permutation statistics was implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. The aim of this paper is to prove that the triple ${\rm (udr, pk, des)}$ is shuffle-compatible as conjectured by Gessel and Zhuang, where ${\rm udr}$ denotes the number of up-down runs, ${\rm pk}$ denotes the peak number, and ${\rm des}$ denotes the descent number. This is accomplished by establishing an ${\rm (udr, pk, des)}$-preserving bijection in the spirit of Baker-Jarvis and Sagan's bijective proofs of shuffle-compatibility property…
Define a permutation to be any sequence of distinct positive integers. Given two permutations p and s on disjoint underlying sets, we denote by p sh s the set of shuffles of p and s (the set of all permutations obtained by interleaving the two permutations). A permutation statistic is a function St whose domain is the set of permutations such that St(p) only depends on the relative order of the elements of p. A permutation statistic is shuffle compatible if the distribution of St on p sh s depends only on St(p) and St(s) and their lengths rather than on the individual permutations themselves. …
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the $\mathrm{fmaj}_k$ indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations which leads to the formula of Chow for signed permutations and the formula of Faliharimalala and Zeng [10] on colored permutations. The third result is an…
Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. We formalize this notion of a shuffle-compatible permutation statistic and introduce the shuffle algebra of a shuffle-compatible permutation statistic, which encodes the distribution of the statistic over shuffles of permutations. This paper develops a theory of shuffle-compatibility for descent statistics (statistics that depend only on the descent set and length) which has close connections to the theory of $P$-partitions, quas…
In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jez, A. Kisielewicz have proved that the direct product of automorphism groups of edge-colored graphs is itself the automorphism groups of an edge-colored graph. In this paper, we study the direct product of two permutation groups such that at least one of them fails to be the automorphism group of an edge-colored graph. We find necessary and sufficient conditions for the direct product to be the automorphism group of an edge-colored graph. The same problem is solved for the edge-colored digraphs.
We generalize the results of Ksavrelof and Zeng about the multidistribution of the excedance number of $S_n$ with some natural parameters to the colored permutation group and to the Coxeter group of type $D$. We define two different orders on these groups which induce two different excedance numbers. Surprisingly, in the case of the colored permutation group, we get the same generalized formulas for both orders.
Consider a permutation p to be any finite list of distinct positive integers. A statistic is a function St whose domain is all permutations. Let S(p,q) be the set of shuffles of two disjoint permutations p and q. We say that St is shuffle compatible if the distribution of St over S(p,q) depends only on St(p), St(q), and the lengths of p and q. This notion is implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. One of the places where shuffles are useful is in describing the product in the algebra of quasisymmetric functions. Recently Adin, Gessel, R…
Given an rc-graph $R$ of permutation $w$ and an rc-graph $Y$ of permutation $v$, we provide an insertion algorithm, which defines an rc-graph $R\leftarrow Y$ in the case when $v$ is a shuffle with the descent at $r$ and $w$ has no descents greater than $r$ or in the case when $v$ is a shuffle, whose shape is a hook. This algorithm gives a combinatorial rule for computing the generalized Littlewood-Richardson coefficients $c^{u}_{wv}$ in the two cases mentioned above.
Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are $γ$-positive (in particular, palindromic and unimodal) polynomials which can be interpreted as $h$-polynomials of certain flag simplicial polytopes and which admit interesting Schur $γ$-positive symmetric function generalizations. This paper introduces analogues of these polynomials for $r$-colored permutations with similar properties and uncovers some new instances of equivariant $γ$-positivity in geometric combinatorics.
We study the generating polynomial of the flag major index with each one-dimensional character, called signed Mahonian polynomial, over the colored permutation group, the wreath product of a cyclic group with the symmetric group. Using the insertion lemma of Han and Haglund-Loehr-Remmel and a signed extension established by Eu et al., we derive the signed Mahonian polynomial over the quotients of parabolic subgroups of the colored permutation group, for a variety of systems of coset representatives in terms of subsequence restrictions. This generalizes the related work over parabolic quotients…
Let a deck of n cards be shuffled by successively exchanging the cards in positions 1, 2, ..., n with cards in randomly chosen positions. We show that for n equal to 18 or greater, the identity permutation is the most likely. We prove a surprising symmetry of the resulting distribution on permutations. We also obtain the limiting distribution of the number of fixed points as n goes to infinity.
We define a new statistic $\mathsf{sor}$ on the set of colored permutations $\mathsf{G}_{r,n}$ and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of $n$ non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: $(\ell,\mathsf{Rmil}^0,\mathsf{Rmil}^1,...,\mathsf{Rmil}^{r-1}$, $\mathsf{Lmil}^0,\mathsf{Lmil}^1,...,\mathsf{Lmil}^{r-1}$, $\mathsf{Lmal}^0,\mathsf{Lmal}^1,...,\mathsf{Lmal}^{r-1}$, $\mathsf{Lmap}^0,\mathsf{Lma…
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…
Using a new colored analogue of P-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algebra. This algebra has a basis consisting of formal sums of colored permutations with the same number of descents (using Steingrímsson's definition of the descent set of a colored permutation). The colored Eulerian descent algebra extends familiar Eulerian descent algebras from the symmetric group algebra and the hyperoctahedral group algebra to colored permutation group algebras. We also describe a set of orthogonal idempotents that span…
We consider several generalizations of the classical $γ$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $γ$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the lite…
We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of decorated rooted trees onto the shuffle algebra. This isomorphism gives a universal way of lifting measure-indexed characters of the Connes-Kreimer algebra into measure-indexed characters of the shuffle algebra, already introduced in \cite{Unterberger} in the framework of rough path theory as the so-called Fourier normal ordering algorithm.
We consider a generalized riffle shuffle on the colored permutation group $G_{p, n}$ and derive a determinantal formula for the probability of finding descents at given positions, proof of which is based on the bijection between the set of shuffles in question and that of non-intersecting lattice paths.
In this paper we introduce the definition of marked permutations. We first present a bijection between Stirling permutations and marked permutations. We then present an involution on Stirling derangements. Furthermore, we present a symmetric bivariate enumerative polynomials on $r$-colored marked permutations. Finally, we give an explanation of $r$-colored marked permutations by using the language of combinatorial objects.
We study sorting by queues that can rearrange their content by applying permutations from a predefined set. These new sorting devices are called shuffle queues and we investigate those of them corresponding to sets of permutations defining some well-known shuffling methods. If $\mathbb{Q}_Σ$ is the shuffle queue corresponding to the shuffling method $Σ$, then we find a number of surprising results related to two natural variations of shuffle queues denoted by $\mathbb{Q}_Σ^{\prime}$ and $\mathbb{Q}_Σ^{\textsf{pop}}$. These require the entire content of the device to be unloaded after a permuta…
We compute the limit distribution for (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In two--colored case our method provides a different proof of a similar result by Tracy and Widom about longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the `colored' problem to the case of usual random permutations using certain combinatorial results and element…
This paper studies statistics of riffle shuffles by relating them to random word statistics with the use of inverse shuffles. Asymptotic normality of the number of descents and inversions in riffle shuffles with convergence rates of order $1/\sqrt{n}$ in the Kolmogorov distance are proven. Results are also given about the lengths of the longest alternating subsequences of random permutations resulting from riffle shuffles. A sketch of how the theory of multisets can be useful for statistics of a variation of top $m$ to random shuffles is presented.
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy all…
This paper studies biased riffle shuffles, first defined by Diaconis, Fill, and Pitman. These shuffles generalize the well-studied Gilbert-Shannon-Reeds shuffle and convolve nicely. An upper bound is given for the time for these shuffles to converge to the uniform distribution; this matches lower bounds of Lalley. A careful version of a bijection of Gessel leads to a generating function for cycle structure after one of these shuffles and gives new results about descents in random permutations. Results are also obtained about the inversion and descent structure of a permutation after one of the…
We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations to obtain our results.
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy all…
We define an excedance number for the multi-colored permutation group, i.e. the wreath product of Z_{r_1} x ... x Z_{r_k} with S_n, and calculate its multi-distribution with some natural parameters. We also compute the multi-distribution of the parameters exc(pi) and fix(pi) over the sets of involutions in the multi-colored permutation group. Using this, we count the number of involutions in this group having a fixed number of excedances and absolute fixed points.
We present methods of calculating statistics generating functions over the colored permutation groups, and generalizing known theorems from the symmetric groups to general colored permutations groups.
We define the excedence set and the excedance word on $G_{r,n}$, generalizing a work of Ehrenborg and Steingrimsson and use the inclusion-exclusion principle to calculate the number of colored permutations having a prescribed excedance word. We show some symmetric properties as Log concavity and unimodality of a specific sequence of excedance words.
We introduce the concept of complete edge-colored permutation graphs as complete graphs that are the edge-disjoint union of "classical" permutation graphs. We show that a graph $G=(V,E)$ is a complete edge-colored permutation graph if and only if each monochromatic subgraph of $G$ is a "classical" permutation graph and $G$ does not contain a triangle with~$3$ different colors. Using the modular decomposition as a framework we demonstrate that complete edge-colored permutation graphs are characterized in terms of their strong prime modules, which induce also complete edge-colored permutation gr…
The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.
We find a combinatorial setting for the coefficients of the Boros-Moll polynomials $P_m(a)$ in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of $P_m(a)$. This approach enables us to give a combinatorial interpretation of the log-concavity of $P_m(a)$ which was conjectured by Moll and confirmed by Kauers and Paule.
It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the flag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley's conjecture, and some combinatorial identities relati…
Recently, Hyatt introduced some colored Eulerian quasisymmetric function to study the joint distribution of excedance number and major index on colored permutation groups. We show how Hyatt's generating function formula for the fixed point colored Eulerian quasisymmetric functions can be deduced from the Decrease value theorem of Foata and Han. Using this generating function formula, we prove two symmetric function generalizations of the Chung-Graham-Knuth symmetrical Eulerian identity for some flag Eulerian quasisymmetric functions, which are specialized to the flag excedance numbers. Combina…
Despite the fact that the field of pattern avoiding permutations has been skyrocketing over the last two decades, there are very few exhaustive generating algorithms for such classes of permutations. In this paper we introduce the notions of regular and colored regular set of forbidden patterns, which are particular cases of right-justified sets of forbidden patterns. We show the (colored) regularity of several sets of forbidden patterns (some of them involving variable length patterns) and we derive a general framework for the efficient generation of permutations avoiding them. The obtained g…
Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after various shuffling methods, emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth correspondence. One consequence is a cycle index for riffle shuffles when one deals from the bottom of the deck; conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. In studying cycle structure of unimodal permutations and their generalizations we are led to an interesting variation of the RSK corresponden…
The number of inversions is a statistic on permutation groups measuring the degree to which the entries of a permutation are out of order. We provide a generalization of that statistic by introducing the statistic number of pseudoinversions on the colored permutation groups. The main motivation to investigate that statistic is the possibility to use it to define a number system and a numeral system on the colored permutation groups. By means of the statistic number of $i$-pseudoinversions, we construct our number system, and a bijection between the set of positive integers and the colored perm…
In this paper generalized Matsumoto-Tits sections lifting permutations to the algebra associated to a generalized virtual braid monoid are defined. They are then applied to study the defining relations of the quantum quasi-shuffle algebras via the total symmetrization operator.
In the cyclic-to-random shuffle, we are given n cards arranged in a circle. At step k, we exchange the k'th card along the circle with a uniformly chosen random card. The problem of determining the mixing time of the cyclic-to-random shuffle was raised by Aldous and Diaconis in 1986. Recently, Mironov used this shuffle as a model for the cryptographic system known as ``RC4'' and proved an upper bound of O(n log n) for the mixing time. We prove a matching lower bound, thus establishing that the mixing time is indeed of order $Θ(n \log n)$. We also prove an upper bound of O(n log n) for the mixi…
Consider a permutation $σ\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $σ_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We study some probabilistic and combinatorial aspects of the shuffle on $S_n$ defined by removing and then randomly reinserting each of the $n$ cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be q…
