scholarly journals Pattern avoidance in inversion sequences

2015 ◽  
Vol 25 (2) ◽  
pp. 157-176 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Generating Functions ◽  
Kernel Method ◽  
Fibonacci Numbers ◽  
Pattern Avoidance ◽  
Explicit Formulas ◽  
Schröder Numbers ◽  
Single Pattern ◽  
Combinatorial Methods

Abstract A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 < ai < i for all z, which is called an inversion sequence. In analogy to the usual case for permutations, the pattern avoidance question is addressed for inversion sequences. In particular, explicit formulas and/or generating functions are derived which count the inversion sequences of a given length that avoid a single pattern of length three. Among the sequences encountered are the Fibonacci numbers, the Schröder numbers, and entry A200753 in OEIS. We make use of both algebraic and combinatorial methods to establish our results. An explicit Injection is given between two of the avoidance classes, and in three cases, the kernel method is used to solve a functional equation satisfied by the generating function enumerating the class in question.

scholarly journals Ascent sequences and Fibonacci numbers

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 703-712
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Functional Equation ◽  
Kernel Method ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Pattern Avoidance ◽  
Combinatorial Structures ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations ◽  
Difficult Cases ◽  
Combinatorial Methods

An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety of other combinatorial structures. Let Fn denote the Fibonacci sequence given by the recurrence Fn = Fn-1 + Fn-2 if n ? 2, with F0 = 0 and F1 = 1. In this paper, we draw connections between ascent sequences and the Fibonacci numbers by showing that several pattern-avoidance classes of ascent sequences are enumerated by either Fn+1 or F2n-1. We make use of both algebraic and combinatorial methods to establish our results. In one of the apparently more difficult cases, we make use of the kernel method to solve a functional equation and thus determine the distribution of some statistics on the avoidance class in question. In two other cases, we adapt the scanning-elements algorithm, a technique which has been used in the enumeration of certain classes of pattern-avoiding permutations, to the comparable problem concerning pattern-avoiding ascent sequences.

scholarly journals Partial matchings and pattern avoidance

2013 ◽  
Vol 7 (1) ◽  
pp. 25-50 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Infinite Family ◽  
Pattern Avoidance ◽  
Explicit Formulas ◽  
Partial Matching ◽  
Natural Notion ◽  
Single Pattern ◽  
Finite Set

A partition of a finite set all of whose blocks have size one or two is called a partial matching. Here, we enumerate classes of partial matchings characterized by the avoidance of a single pattern, specializing a natural notion of partition containment that has been introduced by Sagan. Let vn(?) denote the number of partial matchings of size n which avoid the pattern ?. Among our results, we show that the generating function for the numbers vn(?) is always rational for a certain infinite family of patterns ?. We also provide some general explicit formulas for vn(?) in terms of vn(p), where p is a pattern contained in ?. Finally, we find, with two exceptions, explicit formulas and/or generating functions for the number of partial matchings avoiding any pattern of length at most five.

scholarly journals Counting water cells in bargraphs of compositions and set partitions

2018 ◽  
Vol 12 (2) ◽  
pp. 413-438 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Horizontal Line ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Unit Square ◽  
Joint Distributions ◽  
Water Cell ◽  
First Moment

In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

scholarly journals Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

On sums related to the numerator of generating functions for the kth power of Fibonacci numbers on sums related to the numerator of generating functions for the kth power of Fibonacci numbers

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Pavel Pražák ◽  
Pavel Trojovský
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Similar Type ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Sums Of Products

AbstractNew results about some sums s n(k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Ý, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $$ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $$ are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s n(k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.

Zeta Mahler measures, multiple zeta values and L-values

2015 ◽  
Vol 11 (07) ◽  
pp. 2239-2246
Author(s):  
Yoshitaka Sasaki
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Mahler Measure ◽  
Multiple Zeta Values ◽  
Explicit Formulas ◽  
Multiple Zeta ◽  
Zeta Values

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

scholarly journals String pattern avoidance in generalized non-crossing trees

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Yidong Sun ◽  
Zhiping Wang
Keyword(s):  
Generating Functions ◽  
Inversion Formula ◽  
Pattern Avoidance ◽  
Lagrange Inversion ◽  
Explicit Formulas ◽  
Lagrange Inversion Formula ◽  
Special Cases ◽  
International Audience ◽  
String Pattern

Combinatorics International audience The problem of string pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding string patterns of length one and two are obtained. The Lagrange inversion formula is used to obtain the explicit formulas for some special cases. A bijection is also established between generalized non-crossing trees with special string pattern avoidance and little Schr ̈oder paths.

scholarly journals Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

2018 ◽  
Vol 27 (1) ◽  
pp. 32-61
Author(s):  
David Callan ◽  
Toufik Mansour
Keyword(s):  
Functional Equation ◽  
Algebraic Equation ◽  
Generating Function ◽  
Recurrence Relations ◽  
Kernel Method ◽  
Permutation Patterns ◽  
Multi Index ◽  
Generating Trees

Abstract Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.

scholarly journals Using Homological Duality in Consecutive Pattern Avoidance

10.37236/2005 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Anton Khoroshkin ◽  
Boris Shapiro
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Large Class ◽  
Generating Function ◽  
Generating Functions ◽  
Pattern Avoidance ◽  
Linear Ordinary Differential Equation ◽  
Sufficient Condition ◽  
Direct Algorithm ◽  
Polynomial Coefficients

Using an approach suggested by Dotsenko and Khoroshkin we present a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows us to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.

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