A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [Amigo, Elizalde, Kennel, J. Combin. Theory Ser. A 115 (2008) 485-504] that the shortest forbidden patterns of the shift on N symbols have length N+2. In this paper we give a characterization of the set of permutations that are realized by the shift on N symbols, and we enumerate them according to their length.
The periodic (ordinal) patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial characterization of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain cyclic permutation associated to the pattern. Signed shifts are an important family of one-dimensional dynamical systems that includes shift maps and the tent map as particular cases. Defined as a function on the set of infinite words on a finite alphabet, a signed shift deletes the first letter and, depending on …
Elizalde (2011) characterized which permutations can be obtained by ordering consecutive elements in the trajectories of (positive) beta-transformations and beta-shifts. We prove similar results for negative bases beta.
Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type $B$: Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableau…
Given a real number beta>1, a permutation pi of length n is realized by the beta-shift if there is some x in [0,1] such that the relative order of the sequence x,f(x),...,f^{n-1}(x), where f(x) is the factional part of beta*x, is the same as that of the entries of pi. Widely studied from such diverse fields as number theory and automata theory, beta-shifts are prototypical examples one-dimensional chaotic dynamical systems. When beta is an integer, permutations realized by shifts where studied in [SIAM J. Discrete Math. 23 (2009), 765-786]. In this paper we generalize some of the results to ar…
A number system coding for the permutations generated by cyclic shift is described. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code describing the symmetry properties of the set of permutations generated by cyclic shift. This code is conjectured to be a combinatorial Gray code listing the set of permutations: this corresponds to an Hamiltonian path of minimal weight in an appropriate regular digraph.
In this note we continue the analysis of permutations that avoid substrings j(j+k), 1 <= j <= n-k, k < n, as well as substrings j(j+k) (mod n), 1 <= j <= n. In the first case the number of such permutations can be obtained from recursions starting from derangement numbers, while in the (mod n) case the number of permutations depends on whether n is prime, and more generally, on whether n and k are relatively prime.
Signed shifts are generalizations of the shift map in which, interpreted as a map from the unit interval to itself sending x to the fractional part of Nx, some slopes are allowed to be negative. Permutations realized by the relative order of the elements in the orbits of these maps have been studied recently by Amigo, Archer and Elizalde. In this paper, we give a complete characterization of the permutations (also called patterns) realized by signed shifts. In the case of the negative shift, which is the signed shift having only negative slopes, we use the characterization to give an exact enu…
The $β$-shift is the transformation from the unit interval to itself that maps $x$ to the fractional part of $βx$. Permutations realized by the relative order of the elements in the orbits of these maps have been studied for positive integer values of $β$ and for real values $β>1$. In both cases, a combinatorial description of the smallest positive value of $β$ needed to realize a permutation is provided. In this paper we extend these results to the case of negative $β$, both in the integer and in the real case. Negative $β$-shifts are related to digital expansions with negative real bases, st…
For a random binary noncoalescing feedback shift register of width $n$, with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$, converges in distribution to the same Poisson-Dirichlet limit as holds for random permutations in $\mathcal{S}_N$, with all $N!$ possible permutations equally likely. Such behavior was conjectured by Golomb, Welch, and Goldstein in 1959.
In this paper, we give bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we get the enumeration of 4123-avoiding up-down alternating permutations of even and odd length by presenting bijections between 4123-avoiding up-down alternating permutations and standard Young tableaux.
We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A_{2n}(2143) of alternating permutations of length 2n avoiding 2143 and standard Young tableaux of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux of shape (n + 2, n + 1, n). We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizati…
Let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k,k+1,\cdots, k+l-1\}$ appear in a set of $l$ consecutive positions. Let $p=\{p_j\}_{j=1}^\infty$ be a distribution on $\mathbb{N}$ with $p_j>0$. Let $P_n$ denote the probability measure on $S_n$ corresponding to the $p$-shifted random permutation. Our main result, under the additional assumption that $\{p_j\}_{j=1}^\infty$ is non-increasing, is that $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k})=\big(\prod_{j=1}^{k-1}\sum_{i=1}^jp_i\big) \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\…
The allowed patterns of a map are those permutations in the same relative order as the initial segments of orbits realized by the map. In this paper, we characterize and provide enumerative bounds for the allowed patterns of signed shifts, a family of maps on infinite words.
We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Ou…
In this paper, we study merging-free partitions with their canonical forms and run-sorted permutations. We give a combinatorial proof of the conjecture made by Nabawanda et al. We describe the distribution of the statistics of runs and right-to-left minima over the set of run-sorted permutations and we give the exponential generating function for their joint distribution. We show the number of right-to-left minima is given by the shifted distribution of the Stirling number of the second kind. We also prove that the non-crossing merging-free partitions are enumerated by powers of 2. We use one …
The allowed patterns of a map on a one-dimensional interval are those permutations that are realized by the relative order of the elements in its orbits. The set of allowed patterns is completely determined by the minimal patterns that are not allowed. These are called basic forbidden patterns. In this paper we study basic forbidden patterns of several functions. We show that the logistic map L_r(x)=rx(1-x) and some generalizations have infinitely many of them for 1<r<=4, and we give a lower bound on the number of basic forbidden patterns of L_4 of each length. Next, we give an upper bound on …
Let $G$ be a graph $G$ whose largest independent set has size $m$. A permutation $π$ of $\{1, \ldots, m\}$ is an {\em independent set permutation} of $G$ if $$ a_{π(1)}(G) \leq a_{π(2)}(G) \leq \cdots \leq a_{π(m)}(G) $$ where $a_k(G)$ is the number of independent sets of size $k$ in $G$. In 1987 Alavi, Malde, Schwenk and Erdős proved that every permutation of $\{1, \ldots, m\}$ is an independent set permutation of some graph with $α(G)=m$, i.e. with largest independent set having size $m$. They raised the question of determining, for each $m$, the smallest number $f(m)$ such that every permut…
Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, as well as the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircas…
We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of "shift-invariant" property (corresponding to the notion of unimodularity for random graphs). We then study two models in the framework of random pattern-avoiding permutations. We compute the local limits of uniform $ρ$-avoiding permutations, for $|ρ|=3,$ when the size of the permutations tends to infinity. The core part of the argument is the desc…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an arbitrary order in a sequence, are in bijection with permutations. The permutation decomposes canonically, by inserting parentheses, into a tree having as nodes a class of permutations which we call primitive. Primitive permutations can be assembled from very simple data. The data for the trees into which a permutation decomposes can be written in a form si…
The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal model…
An infinite permutation can be defined as a linear ordering of the set of natural numbers. In particular, an infinite permutation can be constructed with an aperiodic infinite word over $\{0,\ldots,q-1\}$ as the lexicographic order of the shifts of the word. In this paper, we discuss the question if an infinite permutation defined this way admits a canonical representative, that is, can be defined by a sequence of numbers from [0, 1], such that the frequency of its elements in any interval is equal to the length of that interval. We show that a canonical representative exists if and only if th…
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain four classical Gray codes for permutations, bitstrings, binary trees and set partitions as special cases. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different fami…
We give some optimal size generating sets for the group generated by shifts and local permutations on the binary full shift. We show that a single generator, namely the fully asynchronous application of the elementary cellular automaton 57 (or, by symmetry, ECA 99), suffices in addition to the shift. In the terminology of logical gates, we have a single reversible gate whose shifts generate all (finitary) reversible gates on infinitely many binary-valued wires that lie in a row and cannot (a priori) be rearranged. We classify pairs of words $u, v$ such that the gate swapping these two words, t…
A permutation $\boldsymbol w$ gives rise to a graph $G_{\boldsymbol w}$; the vertices of $G_{\boldsymbol w}$ are the letters in the permutation and the edges of $G_{\boldsymbol w}$ are the inversions of $\boldsymbol w$. We find that the number of trees among permutation graphs with $n$ vertices is $2^{n-2}$ for $n\ge 2$. We then study $T_n$, a uniformly random tree from this set of trees. In particular, we study the number of vertices of a given degree in $T_n$, the maximum degree in $T_n$, the diameter of $T_n$, and the domination number of $T_n$. Denoting the number of degree-$k$ vertices in…
It is shown that shift graphs can be realized as disjointness graphs of 1-intersecting curves in the plane. This implies that the latter class of graphs is not $χ$-bounded.
Observers in different inertial frames can see a set of spacelike separated events as occurring in different orders. Various restrictions are studied on the possible orderings of events that can be observed. In 1+1-dimensional spacetime {(1 2 3), (2 3 1), (3 1 2)} is a disallowed set of permutations. In 3+1-dimensional spacetime, any four different permutations on the ordering of n events can be seen by four different observers, and there is a set of four events such that any of the 4!=24 possible orderings can be observed in some inertial reference frame. A more complicated problem is that of…
We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. In particular, we are interested in the number of classes created in $S_n$ by each relation and in characterizing these classes. Imposing the condition that the partition o…
The stack sort algorithm has been the subject of extensive study over the years. In this paper we explore a generalized version of this algorithm where instead of avoiding a single decrease, the stack avoids a set $T$ of permutations. We let $s_T$ denote this map. We classify for which sets $T$ the map $s_T$ is bijective. A corollary to this answers a question of Baril, Cerbai, Khalil, and Vajnovszki about stack sort composed with $s_{\{σ,τ\}}$, known as the $(σ,τ)$-machine. This fully classifies for which $σ$ and $τ$ the preimage of the identity under the $(σ,τ)$-machine is counted by the Cat…
