(Un)ranking Permutation Classes
Permutations avoiding a pattern of length three are enumerated by the Catalan numbers. In this work, we present methods for ranking and unranking such permutations in lexicographic or colexicographic order.
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arXiv subjects
Vincent Vajnovszki
Publications and source records attributed to Vincent Vajnovszki.
Greedy Gray Codes for some Restricted Classes of Binary Words
We investigate the existence of greedy Gray codes, based on the choice of the first element in the code, for two classes of binary words: generalized Fibonacci words and generalized Dyck words.
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Enumeration of Dyck paths with air pockets
We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the distribution of patterns as peaks, returns and pyramids. Then, we deduce the popularities and asymptotic expectations of these patterns and point out a link between the popularity of pyramids and a special kind of closed smooth self-overlapping curves, a subset of Fibonacci meanders. A similar study is conducted for non-decreasing Dyck paths with air pockets.
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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.
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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.
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Gray codes for Fibonacci q-decreasing words
An $n$-length binary word is $q$-decreasing, $q\geq 1$, if every of its length maximal factor of the form $0^a1^b$ satisfies $a=0$ or $q\cdot a > b$.We show constructively that these words are in bijection with binary words having no occurrences of $1^{q+1}$, and thus they are enumerated by the $(q+1)$-generalized Fibonacci numbers. We give some enumerative results and reveal similarities between $q$-decreasing words and binary words having no occurrences of $1^{q+1}$ in terms of frequency of $1$ bit. In the second part of our paper, we provide an efficient exhaustive generating algorithm for $q$-decreasing words in lexicographic order, for any $q\geq 1$, show the existence of 3-Gray codes and explain how a generating algorithm for these Gray codes can be obtained. Moreover, we give the construction of a more restrictive 1-Gray code for $1$-decreasing words, which in particular settles a conjecture stated recently in the context of interconnection networks by Eğecioğlu and Iršič.
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The equidistribution of some Mahonian statistics over permutations avoiding a pattern of length three
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 foze" 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, foze" is one of the statistics defined in terms of generalized patterns in the 2000 pioneering paper of Babson and Steingrímsson, and makl is one of the statistics defined by Clarke, Steingrímsson and Zeng in 1997. These results solve several conjectures posed by Amini in 2018.
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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 known, and thus the corresponding results establish new combinatorial interpretations for them.
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Popularity of patterns over $d$-equivalence classes of words and permutations
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.
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On shortening u-cycles and u-words for permutations
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 $k=0,1,\ldots,(n-2)!$.
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Exhaustive generation for permutations avoiding a (colored) regular sets of patterns
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 the ECO method introduced in the context of enumeration and random generation of combinatorial objects by Barcucci et al. in 1999, and developed later by Bacchelli et al. in 2004, for instance. For some classes of permutations falling under our general framework, the corresponding counting sequences are classical in combinatorics, such as Pell, Fibonacci, Catalan, Schröder and binomial transform of Padovan sequence.
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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 Encyclopedia of Integer Sequences.
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Mahonian STAT on rearrangement class of words
In 2000, Babson and Steingrímsson 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 joint equidistribution between two sextuple statistics over $R(w)$. Our involution builds on Burstein's involution and Foata-Schützenberger's involution that utilizes the celebrated RSK algorithm.
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More restricted growth functions: Gray codes and exhaustive generations
A Gray code for a combinatorial class is a method for listing the objects in the class so that successive objects differ in some prespecified, small way, typically expressed as a bounded Hamming distance. In a previous work, the authors of the present paper showed, among other things, that the $m$-ary Reflected Gray Code Order yields a Gray code for the set of restricted growth functions. Here we further investigate variations of this order relation, and give the first Gray codes and efficient generating algorithms for bounded restricted growth functions.
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Patterns in treeshelves
We study the distribution and the popularity of left children on sets of treeshelves avoiding a pattern of size three. (Treeshelves are ordered binary increasing trees where every child is connected to its parent by a left or a right link.) The considered patterns are sub-treeshelves, and for each such a pattern we provide exponential generating function for the corresponding distribution and popularity. Finally, we present constructive bijections between treeshelves avoiding a pattern of size three and some classes of simpler combinatorial objects.
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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 Möbius 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 and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give a bijective proof of Visontai's conjecture.
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Gray coding planar maps
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.
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The equidistribution of some length three vincular patterns on $S_n(132)$
In 2012 Bóna showed the rather surprising fact that the cumulative number of occurrences of the classical patterns $231$ and $213$ are the same on the set of permutations avoiding $132$, beside the pattern based statistics $231$ and $213$ do not have the same distribution on this set. Here we show that if it is required for the symbols playing the role of $1$ and $3$ in the occurrences of $231$ and $213$ to be adjacent, then the obtained statistics are equidistributed on the set of $132$-avoiding permutations. Actually, expressed in terms of vincular patterns, we prove the following more general results: the statistics based on the patterns $b-ca$, $b-ac$ and $ba-c$, together with other statistics, have the same joint distribution on $S_n(132)$, and so do the patterns $bc-a$ and $c-ab$; and up to trivial transformations, these statistics are the only based on length three proper (not classical nor adjacent) vincular patterns which are equidistributed on a set of permutations avoiding a classical length three pattern.
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