scholarly journals On Growth Rates of Permutations, Set Partitions, Ordered Graphs and Other Objects

10.37236/799 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Martin Klazar
Keyword(s):  
Growth Rates ◽  
Fibonacci Numbers ◽  
Complete Graphs ◽  
Linear Orders ◽  
Set Partitions ◽  
Ordered Graphs ◽  
Containment Order ◽  
Hereditary Properties ◽  
Finite Linear ◽  
Lower Ideal

For classes ${\cal O}$ of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order $\preceq$ (containment of permutations, subgraph relation etc.), we investigate restrictions on the function $f(n)$ counting objects with size $n$ in a lower ideal in $({\cal O},\preceq)$. We present a framework of edge $P$-colored complete graphs $({\cal C}(P),\preceq)$ which includes many of these situations, and we prove for it two such restrictions (jumps in growth): $f(n)$ is eventually constant or $f(n)\ge n$ for all $n\ge 1$; $f(n)\le n^c$ for all $n\ge 1$ for a constant $c>0$ or $f(n)\ge F_n$ for all $n\ge 1$, $F_n$ being the Fibonacci numbers. This generalizes a fragment of a more detailed theorem of Balogh, Bollobás and Morris on hereditary properties of ordered graphs.

scholarly journals Statistical Distributions and $q$-Analogues of $k$-Fibonacci Numbers

10.37236/2550 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Adam M Goyt ◽  
Brady L Keller ◽  
Jonathan E Rue
Keyword(s):  
Fibonacci Numbers ◽  
Natural Extension ◽  
Statistical Distributions ◽  
Set Partition ◽  
Set Partitions ◽  
Partition Statistics

We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most k.  The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics.  We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers.  This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations.  In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles.  We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.

scholarly journals Hereditary properties of partitions, ordered graphs and ordered hypergraphs

2006 ◽  
Vol 27 (8) ◽  
pp. 1263-1281 ◽  
Author(s):  
József Balogh ◽  
Béla Bollobás ◽  
Robert Morris
Keyword(s):  
Ordered Graphs ◽  
Hereditary Properties

scholarly journals Hereditary Properties of Ordered Graphs

2007 ◽  
pp. 179-213 ◽  
Author(s):  
József Balogh ◽  
Béla Bollobás ◽  
Robert Morris
Keyword(s):  
Ordered Graphs ◽  
Hereditary Properties

scholarly journals On Growth Rates of Closed Permutation Classes

10.37236/1682 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Tomáš Kaiser ◽  
Martin Klazar
Keyword(s):  
Growth Rates ◽  
Polynomial Growth ◽  
Positive Root ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Counting Functions

A class of permutations $\Pi$ is called closed if $\pi\subset\sigma\in\Pi$ implies $\pi\in\Pi$, where the relation $\subset$ is the natural containment of permutations. Let $\Pi_n$ be the set of all permutations of $1,2,\dots,n$ belonging to $\Pi$. We investigate the counting functions $n\mapsto|\Pi_n|$ of closed classes. Our main result says that if $|\Pi_n| < 2^{n-1}$ for at least one $n\ge 1$, then there is a unique $k\ge 1$ such that $F_{n,k}\le |\Pi_n|\le F_{n,k}\cdot n^c$ holds for all $n\ge 1$ with a constant $c>0$. Here $F_{n,k}$ are the generalized Fibonacci numbers which grow like powers of the largest positive root of $x^k-x^{k-1}-\cdots-1$. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.

A Tableau System for Right Propositional Neighborhood Logic over Finite Linear Orders: An Implementation

2013 ◽  
pp. 74-80 ◽  
Author(s):  
Davide Bresolin ◽  
Dario Della Monica ◽  
Angelo Montanari ◽  
Guido Sciavicco
Keyword(s):  
Linear Orders ◽  
Tableau System ◽  
Finite Linear

scholarly journals Testing Hereditary Properties of Ordered Graphs and Matrices

2017 ◽  
Author(s):  
Noga Alon ◽  
Omri Ben-Eliezer ◽  
Eldar Fischer
Keyword(s):  
Ordered Graphs ◽  
Hereditary Properties

scholarly journals Permutation Statistics and $q$-Fibonacci Numbers

10.37236/190 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Adam M. Goyt ◽  
David Mathisen
Keyword(s):  
Fibonacci Numbers ◽  
Permutation Statistics ◽  
Set Partition ◽  
Set Partitions ◽  
Restricted Permutations ◽  
Mahonian Statistics ◽  
Partition Statistics

In a recent paper, Goyt and Sagan studied distributions of certain set partition statistics over pattern restricted sets of set partitions that were counted by the Fibonacci numbers. Their study produced a class of $q$-Fibonacci numbers, which they related to $q$-Fibonacci numbers studied by Carlitz and Cigler. In this paper we will study the distributions of some Mahonian statistics over pattern restricted sets of permutations. We will give bijective proofs connecting some of our $q$-Fibonacci numbers to those of Carlitz, Cigler, Goyt and Sagan. We encode these permutations as words and use a weight to produce bijective proofs of $q$-Fibonacci identities. Finally, we study the distribution of some of these statistics on pattern restricted permutations that West showed were counted by even Fibonacci numbers.

scholarly journals Pattern Avoidance in Ascent Sequences

10.37236/713 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Paul Duncan ◽  
Einar Steingrímsson
Keyword(s):  
Growth Rates ◽  
Pattern Avoidance ◽  
Combinatorial Structures ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Ascent Sequences ◽  
Wilf Equivalence

Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to $(2+2)$-free posets and various other combinatorial structures. We study pattern avoidance in ascent sequences, giving several results for patterns of lengths up to 4, for Wilf equivalence and for growth rates. We establish bijective connections between pattern avoiding ascent sequences and various other combinatorial objects, in particular with set partitions. We also make a number of conjectures related to all of these aspects.

Reduced growth rates of hair in mice following radiation

1966 ◽  
Vol 94 (4) ◽  
pp. 491-498 ◽  
Author(s):  
F. D. Malkinson
Keyword(s):  
Growth Rates
Sign in / Sign up

Export Citation Format

Share Document