Combinatorial parameters on bargraphs of permutations (original) (raw)
Related papers
Statistics on bargraphs of inversion sequences of permutations
2020
An inversion sequence (x1, . . . , xn) is one such that 1 ≤ xi ≤ i for all 1 ≤ i ≤ n. We first consider the joint distribution of the area and perimeter statistics on the set In of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived from recurrences satisfied by this distribution. Explicit formulas for the generating functions are found in some special cases as are expressions for the totals of the respective statistics on In. A similar treatment is provided for the joint distribution on In for the statistics recording the number of levels, descents and ascents. Some connections are made between specific cases of this latter distribution and the Stirling numbers of the first kind and Eulerian numbers.
Bijective Enumeration and Sign-Imbalance for Permutation Depth and Excedances
Electronic proceedings in theoretical computer science, 2024
We present a simplified variant of Biane's bijection between permutations and 3-colored Motzkin paths with weight that keeps track of the inversion number, excedance number and a statistic socalled depth of a permutation. This generalizes a result by Guay-Paquet and Petersen about a continued fraction of the generating function for depth on the symmetric group S n of permutations. In terms of weighted Motzkin path, we establish an involution on S n that reverses the parities of depth and excedance numbers simultaneously, which proves that the numbers of permutations with even and odd depth (excedance numbers, respectively) are equal if n is even and differ by the tangent number if n is odd. Moreover, we present some interesting sign-imbalance results on permutations and derangements, refined with respect to depth and excedance numbers.
2010
Let π = π 1 π 2 • • • π n be any permutation of length n, we say a descent π i π i+1 is a lower, middle, upper if there exists j > i + 1 such that π j < π i+1 , π i+1 < π j < π i , π i < π j , respectively. Similarly, we say a rise π i π i+1 is a lower, middle, upper if there exists j > i + 1 such that π j < π i , π i < π j < π i+1 , π i+1 < π j , respectively. In this paper we give an explicit formula for the generating function for the number of permutations of length n according to number of upper, middle, lower rises, and upper, middle, lower descents. This allows us to recover several known results in the combinatorics of permutation patterns as well as many new results. For example, we give an explicit formula for the generating function for the number of permutations of length n having exactly m middle descents.
Equivalence of the descents statistic on some (4,4)-avoidance classes of permutations
Journal of Difference Equations and Applications
In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the n-th large Schröder number and thus one obtains a new bivariate refinement of these numbers as a consequence. We make use of auxiliary combinatorial statistics, special generating functions (specific to each class) and the kernel method to establish our results. In some cases, we utilize the conjecture itself in a creative way to aid in solving the system of functional equations satisfied by the associated generating functions.
The Descent Statistic on Signed Simsun Permutations
J. Autom. Lang. Comb., 2017
In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.
Counting permutations by alternating descents
The Electronic Journal of Combinatorics, 2014
We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of David and Barton's generating function for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel.
The Descent Statistic on 123-AVOIDING Permutations
2010
We exploit Krattenthaler's bijection between 123-avoi- ding permutations and Dyck paths to determine the Eulerian dis- tribution over the set Sn(123) of 123-avoiding permutations in Sn. In particular, we show that the descents of a permutation corre- spond to valleys and triple ascents of the associated Dyck path. We get the Eulerian numbers of Sn(123) by studying the joint distribution
2019
In this review, we present the main results related to bargraphs from the enumerative point of view. We consider several geometrically motivated statistics in combinatorial families: compositions, words, set partitions, permutations and integer partitions, when such families are presented as bargraphs. The review contains presentation of main results of each considered paper. The main discussion is preceded by short historical notes. Methods used throughout the review are discussed briefly. The review includes an up-to-date bibliography.
The descent statistic over 123-avoiding permutations
2009
We exploit Krattenthaler’s bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set Sn(123) of 123-avoiding permutations in Sn. In particular, we show that the descents of a permutation correspond to valleys and triple falls of the associated Dyck path. We get the Eulerian numbers of Sn(123) by studying the joint distribution of these two statistics on Dyck paths.