A permutation code preserving a double Eulerian bistatistic

Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. Among the techniques used by Aas are the M\"obius inversion formula and isomorphism of labeled rooted trees. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two $5$-tuples of set-valued statistics on the set of permutations an…

Efficient generation of restricted growth words

A length n restricted growth word is a word w=w"1w"2...w"n over the set of integers where w"1=0 and each w"i, i>1, lies between 0 and the value of a word statistics of the prefix w"1w"2...w"i"-"1 of w, plus one. Restricted growth words simultaneously generalize combinatorial objects as restricted growth functions, staircase words and ascent or binary sequences. Here we give a generic generating algorithm for restricted growth words. It produces a Gray code and runs in constant average time provided that the corresponding statistics has some local properties.

The equidistribution of some Mahonian statistics over permutations avoiding a pattern of length three

Abstract We prove the equidistribution of several multistatistics over some classes of permutations avoiding a 3-length pattern. We deduce the equidistribution, on the one hand of inv and foz e ″ statistics, and on the other hand that of maj and makl statistics, over these classes of pattern avoiding permutations. Here inv and maj are the celebrated Mahonian statistics, foz e ″ is one of the statistics defined in terms of generalized patterns in the 2000 pioneering paper of Babson and Steingrimsson, and makl is one of the statistics defined by Clarke, Steingrimsson and Zeng in (1997) [5] . These results solve several conjectures posed by Amini in (2018) [1] .

A trace partitioned Gray code forq-ary generalized Fibonacci strings

AbstractWe provide a trace partitioned Gray code for the set of q-ary strings avoiding a pattern constituted by k consecutive equal symbols. The definition of this Gray code is based on two different constructions, according to the parity of q. This result generalizes, and is based on, a Gray code for binary strings avoiding k consecutive 0's.

Popularity of patterns over $d$-equivalence classes of words and permutations

Abstract Two same length words are d-equivalent if they have same descent set and same underlying alphabet. In particular, two same length permutations are d-equivalent if they have same descent set. The popularity of a pattern in a set of words is the overall number of copies of the pattern within the words of the set. We show the far-from-trivial fact that two patterns are d-equivalent if and only if they are equipopular over any d-equivalence class, and this equipopularity does not follow obviously from a trivial equidistribution.

Catalan and Schröder permutations sortable by two restricted stacks

Abstract Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the…

Restricted 123-avoiding Baxter permutations and the Padovan numbers

AbstractBaxter studied a particular class of permutations by considering fixed points of the composite of commuting functions. This class is called Baxter permutations. In this paper we investigate the number of 123-avoiding Baxter permutations of length n that also avoid (or contain a prescribed number of occurrences of) another certain pattern of length k. In several interesting cases the generating function depends only on k and is expressed via the generating function for the Padovan numbers.

Pattern-based permutation statistics and code transforms

International audience

Descent distribution on Catalan words avoiding a pattern of length at most three

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of descents on this set. Some of the obtained enumerating sequences are not yet recorded in the On-line En…

Exhaustive generation for permutations avoiding (colored) regular sets of patterns

Abstract 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 generating algorithms are based on succession functions, a notion which is a byproduct of t…

Generalized Schröder permutations

We give the generating function for the integer sequence enumerating a class of pattern avoiding permutations depending on two parameters: m and p. The avoided patterns are the permutations of length m with the largest element in the first position and the second largest in one of the last p positions. For particular instances of m and p we obtain pattern avoiding classes enumerated by Schroder, Catalan and central binomial coefficient numbers, and thus, the obtained two-parameter generating function gathers under one roof known generating functions and expresses new ones. This work generalizes some earlier results of Barcucci et al. (2000) [2], Kremer (2000) [5] and Kremer (2003) [6].

Ranking and unrankingk-ary trees with a 4k –4 letter alphabet

Abstract The problem of the direct generation in A-order of binary trees was stated by Zaks in 1980. In 1988 Roelants van Baronaigien and Ruskey gave a solution for k-ary trees with n internal nodes using an encoding sequence of kn+1 integers between 1 and n. Vajnovszki and Pallo improved this result for binary trees in 1994 using words of length n–1 on a four letter alphabet. Recently Korsh generalized the Vajnovszki and Pallo’s generating algorithm to k-ary trees using an alphabet whose cardinality depends on k but not on n. We give in this paper ranking and unranking algorithms for k-ary trees using the Korsh’s encoding scheme.

Minimal change list for Lucas strings and some graph theoretic consequences

AbstractWe give a minimal change list for the set of order p length-n Lucas strings, i.e., the set of length-n binary strings with no p consecutive 1's nor a 1ℓ prefix and a 1m suffix with ℓ+m⩾p. The construction of this list proves also that the order p n-dimensional Lucas cube has a Hamiltonian path if and only if n is not a multiple of p+1, and its second power always has a Hamiltonian path.

Equipopularity of descent-equivalent patterns over descent-equivalence classes of words and permutations

More restrictive Gray codes for necklaces and Lyndon words

In the last years, the order induced by the Binary Reflected Gray Code or its generalizations shown an increasing interest. In this note we show that the BRGC order induces a cyclic 2-Gray code on the set of binary necklaces and Lyndon words and a cyclic 3-Gray code on the unordered counterparts. This is an improvement and a generalization to unlabeled words of the result in [V. Vajnovszki, Gray code order for Lyndon words, Discrete Math. Theoret. Comput. Sci. 9 (2) (2007) 145-152; M. Weston, V. Vajnovszki, Gray codes for necklaces and Lyndon words of arbitrary base, Pure Mathematics and Applications/Algebra and Theoretical Computer Science, in press]; however an algorithmic implementation …

Combinatorial Gray codes for classes of pattern avoiding permutations

The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, Schr\"oder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray codes for $\s_n(\tau)$ for all $\tau\in \s_3$ and the obtained Gray codes have distances 4 and 5.

On shortening u-cycles and u-words for permutations

Abstract 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 , … , ( n − 2 ) ! .

Gray visiting Motzkins

We present the first Gray code for Motzkin words and their generalizations: k colored Motzkin words and Schroder words. The construction of these Gray codes is based on the observation that a k colored Motzkin word is the shuffle of a Dyck word by a k-ary variation on a trajectory which is a combination. In the final part of the paper we give some algorithmic considerations and other possible applications of the techniques introduced here.

Catalan words avoiding pairs of length three patterns

Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are kn…

On the loopless generation of binary tree sequences

Weight sequences were introduced by Pallo in 1986 for coding binary trees and he presented a constant amortized time algorithm for their generation in lexicographic order. A year later, Roelants van Baronaigien and Ruskey developed a recursive constant amortized time algorithm for generating Gray code for binary trees in Pallo's representation. It is common practice to find a loopless generating algorithm for a combinatorial object when enunciating a Gray code for this object. In this paper we regard weight sequences as variations and apply a Williamson algorithm in order to obtain a loopless generating algorithm for the Roelants van Baronaigien and Ruskey's Gray code for weight sequences.

Two Reflected Gray Code-Based Orders on Some Restricted Growth Sequences

We consider two order relations: that induced by the m-ary reflected Gray code and a suffix partitioned variation of it. We show that both of them when applied to some sets of restricted growth sequences still yield Gray codes. These sets of sequences are: subexcedant and ascent sequences, restricted growth functions and staircase words. In particular, we give the first suffix partitioned Gray codes for restricted growth f unctions and ascent sequences; these latter sequences code various combinatorial classes as interval orders, upper triangular matrices without zero rows and zero columns whose non-negative integer entries sum up to n, and certain pattern-avoiding permutations. For each Gr…

Restricted compositions and permutations: from old to new Gray codes

Any Gray code for a set of combinatorial objects defines a total order relation on this set: x is less than y if and only if y occurs after x in the Gray code list. Let @? denote the order relation induced by the classical Gray code for the product set (the natural extension of the Binary Reflected Gray Code to k-ary tuples). The restriction of @? to the set of compositions and bounded compositions gives known Gray codes for those sets. Here we show that @? restricted to the set of bounded compositions of an interval yields still a Gray code. An n-composition of an interval is an n-tuple of integers whose sum lies between two integers; and the set of bounded n-compositions of an interval si…

Right-Justified Characterization for Generating Regular Pattern Avoiding Permutations

ECO-method and its corresponding succession rules allow to recursively define and construct combinatorial objects. The induced generating trees can be coded by corresponding pattern avoiding permutations. We refine succession rules by using succession functions in case when avoided patterns are regular or c-regular. Although regular patterns are hard to be recognized in general, we give a characterization for its right-justified property which is a prerequisite in the definition of the regular pattern. Based on this characterization, we show the (c-)regularity for various classes of permutations avoiding sets of patterns with variable lengths. Last, the technique of succession functions per…

Parallel Algorithms for Listing Well-Formed Parentheses Strings

We present two cost-optimal parallel algorithms generating the set of all well-formed parentheses strings of length 2n with constant delay for each generated string. In our first algorithm we generate in lexicographic order well-formed parentheses strings represented by bitstrings, and in the second one we use the representation by weight sequences. In both cases the computational model is based on an architecture CREW PRAM, where each processor performs the same algorithm simultaneously on a different set of data. Different processors can access the shared memory at the same time to read different data in the same or different memory locations, but no two processors are allowed to write i…

Combinatorial isomorphism between Fibonacci classes

Abstract In 1985 Simion and Schmidt showed that the set S n (T 3) of length n permutations avoiding the set of patterns T 3={123, 132, 213} is counted by (the second order) Fibonacci numbers. They also presented a constructive bijection between the set F n–1 of length (n–1) binary strings with no two consecutive 1s and S n (T 3). In 2005, Egge and Mansour generalized the first Simion-Simion’s result and showed that S n (T p ), the set of permutations avoiding the patterns T p ={12…p, 132, 213}, is counted by the (p–1)th order Fibonacci numbers. In this paper we extend the second Simion-Schmidt’s result by giving a bijection between the set of length (n–1) binary strings with no (p–1) consec…

A new Euler–Mahonian constructive bijection

AbstractUsing generating functions, MacMahon proved in 1916 the remarkable fact that the major index has the same distribution as the inversion number for multiset permutations, and in 1968 Foata gave a constructive bijection proving MacMahon’s result. Since then, many refinements have been derived, consisting of adding new constraints or new statistics.Here we give a new simple constructive bijection between the set of permutations with a given number of inversions and those with a given major index. We introduce a new statistic, mix, related to the Lehmer code, and using our new bijection we show that the bistatistic (mix,INV) is Euler–Mahonian. Finally, we introduce the McMahon code for …

Generating restricted classes of involutions, Bell and Stirling permutations

AbstractWe present a recursive generating algorithm for unrestricted permutations which is based on both the decomposition of a permutation as a product of transpositions and that as a union of disjoint cycles. It generates permutations at each recursive step and slight modifications of it produce generating algorithms for Bell permutations and involutions. Further refinements yield algorithms for these classes of permutations subject to additional restrictions: a given number of cycles or/and fixed points. We obtain, as particular cases, generating algorithms for permutations counted by the Stirling numbers of the first and second kind, even permutations, fixed-point-free involutions and d…

SYSTOLIC GENERATION OF k-ARY TREES

The only parallel generating algorithms for k-ary trees are those of Akl and Stojmenović in 1996 and of Vajnovszki and Phillips in 1997. In the first of them, trees are represented by an inversion table and the processor model is a linear aray multicomputer. In the second, trees are represented by bitstrings and the algorithm executes on a shared memory multiprocessor. In this paper we give a parallel generating algorithm for k-ary trees represented by generalized P–sequences for execution on a linear array multicomputer.

Some Generalizations of a Simion Schmidt Bijection

In 1985, Simion and Schmidt gave a constructive bijection φ from the set of all length (n-1) binary strings having no two consecutive 1s to the set of all length n permutations avoiding all patterns in {123,132,213}. In this paper, we generalize φ to an injective function from {0,1}n-1 to the set Sn of all length n permutations and derive from it four bijections φ : P →Q where P⊆{0,1}n-1 and Q ⊂ Sn. The domains are sets of restricted binary strings and the codomains are sets of pattern-avoiding permutations. As a particular case we retrieve the original Simion–Schmidt bijection. We also show that the bijections obtained are actually combinatorial isomorphisms, i.e. closeness-preserving bije…

Gray code for derangements

AbstractWe give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no fixed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of fixed points bounded between two constants.

Grand Dyck paths with air pockets

Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths with air pockets by allowing them to go below the $x$-axis. We present enumerative results on GDAP (or their prefixes) subject to various restrictions such as maximal/minimal height, ordinate of the last point and particular first return decomposition. In some special cases we give bijections with other known combinatorial classes.

A new vincular pattern based Mahonian statistic on words

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An efficient Gray code algorithm for generating all permutations with a given major index

Abstract In Effler and Ruskey (2003) [1] the authors give an algorithm, which appears to be CAT, for generating permutations with a given major index. In the present paper we give a new algorithm for generating a Gray code for subexcedant sequences. We show that this algorithm is CAT and modify it into a CAT generating algorithm for a Gray code for permutations with a given major index.

A loopless algorithm for generating the permutations of a multiset

AbstractMany combinatorial structures can be constructed from simpler components. For example, a permutation can be constructed from cycles, or a Motzkin word from a Dyck word and a combination. In this paper we present a constructor for combinatorial structures, called shuffle on trajectories (defined previously in a non-combinatorial context), and we show how this constructor enables us to obtain a new loopless generating algorithm for multiset permutations from similar results for simpler objects.

Lehmer code transforms and Mahonian statistics on permutations

Abstract In 2000 Babson and Steingrimsson introduced the notion of vincular patterns in permutations. They show that essentially all well-known Mahonian permutation statistics can be written as combinations of such patterns. Also, they proved and conjectured that other combinations of vincular patterns are still Mahonian. These conjectures were proved later: by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In this paper we give an alternative proof of some of these results. Our approach is based on permutation codes which, like the Lehmer code, map bijectively permutations onto subexcedant sequences. More precisely, we give several code transforms (i.e., bijections…

Fibonacci q-decreasing words: enumerative results and Gray codes

Mahonian STAT on rearrangement class of words

In 2000, Babson and Steingr\'{i}msson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on $R(w)$, the rearrangement class of a given word $w$. We construct such an involution in this paper, which yields a comparable jo…

Qubonacci words

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Mahonian STAT on words

In 2000, Babson and Steingrimsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006.In 2010, Burstein refined some of these results by giving a bijection between permutations with a fixed value for the major index and those with the same value for STAT , where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingrimsson's paper. Several other statistics are preserved as well by Burstein's bijection.At…

Asymptotic bit frequency in Fibonacci words

It is known that binary words containing no $k$ consecutive 1s are enumerated by $k$-step Fibonacci numbers. In this note we discuss the expected value of a random bit in a random word of length $n$ having this property.

Gray coding cubic planar maps

International audience; The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize beta-description trees to cyclicly Gray code three classes of cubic planar maps, namely, bicubic planar maps, 3-connected cubic planar maps, and cubic non-separable planar maps. (C) 2015 Elsevier B.V. All rights reserved.

Gray codes for order p Lucas strings

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A Loopless Generation of Bitstrings without p Consecutive Ones

Let F n (p) be the set of all n-length bitstrings such that there are no p consecutive ls. F n (p) is counted with the pth order Fibonacci numbers and it may be regarded as the subsets of {1, 2,…, n} without p consecutive elements and bitstrings in F n (p) code a particular class of trees or compositions of an integer. In this paper we give a Gray code for F n (p) which can be implemented in a recursive generating algorithm, and finally in a loopless generating algorithm.

Loop-free Gray code algorithm for the e-restricted growth functions

The subject of Gray codes algorithms for the set partitions of {1,2,...,n} had been covered in several works. The first Gray code for that set was introduced by Knuth (1975) [5], later, Ruskey presented a modified version of [email protected]?s algorithm with distance two, Ehrlich (1973) [3] introduced a loop-free algorithm for the set of partitions of {1,2,...,n}, Ruskey and Savage (1994) [9] generalized [email protected]?s results and give two Gray codes for the set of partitions of {1,2,...,n}, and recently, Mansour et al. (2008) [7] gave another Gray code and loop-free generating algorithm for that set by adopting plane tree techniques. In this paper, we introduce the set of e-restricte…

A loop-free two-close Gray-code algorithm for listing k-ary Dyck words

AbstractP. Chase and F. Ruskey each published a Gray code for length n binary strings with m occurrences of 1, coding m-combinations of n objects, which is two-close—that is, in passing from one binary string to its successor a single 1 exchanges positions with a 0 which is either adjacent to the 1 or separated from it by a single 0. If we impose the restriction that any suffix of a string contains at least k−1 times as many 0's as 1's, we obtain k-suffixes: suffixes of k-ary Dyck words. Combinations are retrieved as special case by setting k=1 and k-ary Dyck words are retrieved as a special case by imposing the additional condition that the entire string has exactly k−1 times as many 0's a…