Permutations Avoiding Certain Partially-Ordered Patterns

A permutation $\pi$ contains a pattern $\sigma$ if and only if there is a subsequence in $\pi$ with its letters in the same relative order as those in $\sigma$. Partially ordered patterns (POPs) provide a convenient way to denote patterns in which the relative order of some of the letters does not matter. This paper elucidates connections between the avoidance sets of a few POPs with other combinatorial objects, directly answering five open questions posed by Gao and Kitaev in 2019. This was done by thoroughly analysing the avoidance sets and developing recursive algorithms to derive these sets and their corresponding combinatorial objects in parallel, which yielded natural bijections. We also analysed an avoidance set whose simple permutations are enumerated by the Fibonacci numbers and derived an algorithm to obtain them recursively.

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A SURVEY OF FACTORIZATION COUNTING FUNCTIONS

International Journal of Number Theory ◽

10.1142/s1793042105000315 ◽

2005 ◽

Vol 01 (04) ◽

pp. 563-581 ◽

Cited By ~ 11

Author(s):

A. KNOPFMACHER ◽

M. E. MAYS

Keyword(s):

Number Theory ◽

Positive Integer ◽

Additive Number Theory ◽

Recursive Algorithms ◽

Additive Number ◽

Combinatorial Objects ◽

Survey Techniques ◽

Counting Functions ◽

General Field ◽

Positive Integers

The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.

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Edgewise Cohen-Macaulay connectivity of partially ordered sets

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-97270 ◽

2018 ◽

Vol 122 (1) ◽

pp. 5

Author(s):

Christos A. Athanasiadis ◽

Myrto Kallipoliti

Keyword(s):

Closed Interval ◽

Convex Polytopes ◽

Partially Ordered Sets ◽

Strong Form ◽

Hasse Diagram ◽

Ordered Sets ◽

Partially Ordered ◽

Open Questions

The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen-Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen-Macaulay poset of the same rank. A corresponding notion of edgewise Cohen-Macaulay connectivity for partially ordered sets is investigated. Examples and open questions are discussed.

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On Partially Ordered Patterns of Length 4 and 5 in Permutations

The Electronic Journal of Combinatorics ◽

10.37236/8605 ◽

2019 ◽

Vol 26 (3) ◽

Author(s):

Alice L. L. Gao ◽

Sergey Kitaev

Keyword(s):

Partially Ordered Set ◽

Ordered Set ◽

Relative Order ◽

Permutation Patterns ◽

Integer Sequences ◽

Arbitrary Length ◽

Partially Ordered ◽

Long Line ◽

Online Encyclopedia ◽

Wilf Equivalence

Partially ordered patterns (POPs) generalize the notion of classical patterns studied widely in the literature in the context of permutations, words, compositions and partitions. In an occurrence of a POP, the relative order of some of the elements is not important. Thus, any POP of length $k$ is defined by a partially ordered set on $k$ elements, and classical patterns correspond to $k$-element chains. The notion of a POP provides a convenient language to deal with larger sets of permutation patterns. This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain 13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

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New families of special numbers for computing negative order Euler numbers and related numbers and polynomials

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm1801001s ◽

2018 ◽

Vol 12 (1) ◽

pp. 1-35 ◽

Cited By ~ 16

Author(s):

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Bernstein Polynomials ◽

Fibonacci Numbers ◽

Stirling Numbers ◽

Euler Numbers ◽

Open Questions ◽

Negative Order ◽

Central Factorial Numbers ◽

Probability And Statistics

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

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On intervals of the consecutive pattern poset

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6380 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Sergi Elizalde ◽

Peter R. W. McNamara

Keyword(s):

Partially Ordered Set ◽

Ordered Set ◽

Relative Order ◽

Partially Ordered ◽

International Audience

International audience The consecutive pattern poset is the infinite partially ordered set of all permutations where σ ≤ τ if τ has a subsequence of adjacent entries in the same relative order as the entries of σ. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have Mo ̈bius function equal to zero.

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On the Rank Function of a Differential Poset

The Electronic Journal of Combinatorics ◽

10.37236/2258 ◽

2012 ◽

Vol 19 (2) ◽

Cited By ~ 3

Author(s):

Richard P. Stanley ◽

Fabrizio Zanello

Keyword(s):

Asymptotic Formula ◽

Projective Planes ◽

Rank Function ◽

Combinatorial Objects ◽

Open Questions ◽

Algebraic Properties ◽

Interval Property ◽

Differential Poset ◽

Special Case ◽

Young’S Lattice

We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions.

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Recursive algorithms in computer science courses: Fibonacci numbers and binomial coefficients

IEEE Transactions on Education ◽

10.1109/13.865200 ◽

2000 ◽

Vol 43 (3) ◽

pp. 273-276 ◽

Cited By ~ 14

Author(s):

I. Stojmenovic

Keyword(s):

Computer Science ◽

Fibonacci Numbers ◽

Binomial Coefficients ◽

Recursive Algorithms ◽

Science Courses ◽

Computer Science Courses

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What’s Next?

Journal of Media Psychology Theories Methods and Applications ◽

10.1027/1864-1105/a000034 ◽

2011 ◽

Vol 23 (1) ◽

pp. 60-63 ◽

Cited By ~ 32

Author(s):

Peter Vorderer

Keyword(s):

Cultural Differences ◽

Factor Model ◽

New Developments ◽

Theoretical Concepts ◽

Open Questions ◽

Entertainment Products ◽

Entertainment Theory

This paper points to new developments in the context of entertainment theory. Starting from a background of well-established theories that have been proposed and elaborated mainly by Zillmann and his collaborators since the 1980s, a new two-factor model of entertainment is introduced. This model encompasses “enjoyment” and “appreciation” as two independent factors. In addition, several open questions regarding cultural differences in humans’ responses to entertainment products or the usefulness of various theoretical concepts like “presence,” “identification,” or “transportation” are also discussed. Finally, the question of why media users are seeking entertainment is brought to the forefront, and a possibly relevant need such as the “search for meaningfulness” is mentioned as a possible major candidate for such an explanation.

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