Quadrant marked mesh patterns in 132-avoiding permutations III

Given a permutation σ = σ1 . . . σn in the symmetric group Sn, we say that σi matches the marked mesh pattern MMP(a, b, c, d) in σ if there are at least a points to the right of σi in σ which are greater than σi, at least b points to the left of σi in σ which are greater than σi, at least c points to the left of σi in σ which are smaller than σi, and at least d points to the right of σi in σ which are smaller than σi. This paper is continuation of the systematic study of the distributions of quad- rant marked mesh patterns in 132-avoiding permutations started in [9] and [10] where we studied the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding permutations where at most two elements of a, b, c, d are greater than zero and the remaining elements are zero. In this paper, we study the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding permutations where at least three of a, b, c, d are greater than zero. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions

Quadrant marked mesh patterns in 132-avoiding permutations I

This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation \sg = \sg_1 ... \sg_n in the symmetric group $S_n$, we say that \sg_i matches the quadrant marked mesh pattern $MMP(a,b,c,d)$ if there are at least $a$ elements to the right of \sg_i in \sg that are greater than \sg_i, at least $b$ elements to left of \sg_i in \sg that are greater than \sg_i, at least $c$ elements to left of \sg_i in \sg that are less than \sg_i, and at least $d$ elements to the right of \sg_i in \sg that are less than \sg_i. We study the distribution of $MMP(a,b,c,d)$ in 132-avoiding permutations. In particular, we study the distribution of $MMP(a,b,c,d)$, where only one of the parameters $a,b,c,d$ are non-zero. In a subsequent paper [7], we will study the the distribution of $MMP(a,b,c,d)$ in 132-avoiding permutations where at least two of the parameters $a,b,c,d$ are non-zero.Comment: Theorem 10 is correcte

The Expected Variation of Random Bounded Integer Sequences of Finite Length

From the enumerative generating function of an abstract adjacency statistic, we deduce the mean and variance of the variation on random permutations, rearrangements, compositions, and bounded integer sequences of finite length

Extending from bijections between marked occurrences of patterns to all occurrences of patterns

We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on $\{1,2,\ldots,2n\}$ to occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results

Computing optimal strategies for a cooperative hat game

We consider a `hat problem' in which each player has a randomly placed stack of black and white hats on their heads, visible to the other player, but not the wearer. Each player must guess a hat position on their head with the goal of both players guessing a white hat. We address the question of finding the optimal strategy, i.e., the one with the highest probability of winning, for this game. We provide an overview of prior work on this question, and describe several strategies that give the best known lower bound on the probability of winning. Upper bounds are also considered here

Consecutive Patterns: From Permutations to Column-Convex Polyominoes and Back

We expose the ties between the consecutive pattern enumeration problems associated with permutations, compositions, column-convex polyominoes, and words. Our perspective allows powerful methods from the contexts of compositions, column-convex polyominoes, and of words to be applied directly to the enumeration of permutations by consecutive patterns. We deduce a host of new consecutive pattern results,including a solution to the (2m+1)-alternating pattern problem on permutations posed by Kitaev

Patterns and statistics on words

We study the enumeration of combinatorial objects by number of occurrences of patterns and other statistics. This work is broken into three main parts. In the first part, we enumerate permutations, compositions, column- convex polyominoes, and words by patterns relating consecutive entries. We show that there is a hierarchy of enumeration problems on these sets of objects, such that the problems in one set may be reformulated in terms of the higher sets, then solved using powerful techniques developed for those sets. We use this viewpoint to solve an open problem due to Kitaev and to produce many extensions of existing results and interesting new results. In the second part, we use the same viewpoint to generalize a theorem due to Garsia and Gessel on the major index statistic. We give many specializations and slight extensions of this result to apply it to a variety of combinatorial objects and variations of the statistic. In the third part, we present general method for finding bijections between sets of objects that preserve various statistics. We use this method to solve problems posed by Claesson and Linusson and by Jones, and we also present several new result

Extending from bijections between marked occurrences of patterns to all occurrences of patterns

Abstract. We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on {1, 2,..., 2n} to occurrences of a certain pattern in permutations in Sn. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results. Résumé. Nous considérons deux dernières problèmes ouverts indiquant que certaines statistiques sur les divers ensembles d’objets combinatoires sont équiréparties. La première, posée par Anders Claesson et Svante Linusson, concerne les imbrications dans des filtrages sur {1, 2,..., 2n} pour les occurrences d’un certain modèle de permutations dans Sn. La seconde, posée par Miles Jones et Jeffrey Remmel, concerne les occurrences d’une large classe de schémas de permutation consécutive aux événements du même modèle dans les cycles de permutations. Nous développons une méthode générale qui résout ces deux problèmes et beaucoup plus. Nous avons également utiliser le principe d’involution Garsia-Milne pour obtenir des preuves purement bijective de ces résultats

On levine's notorious hat puzzle

The Levine hat game requires n players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The players collectively win if and only if all their guesses are correct. In this paper we give an overview of what is known about strategies for this game, including an extended discussion of the case with n = 2 players (and a conjecture for an optimal strategy in this case). We also prove that Vn, the optimal value of the joint success probability in the n-player game, is a strictly decreasing function of n.SCOPUS: ar.jinfo:eu-repo/semantics/publishe