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Counting Water Cells in Pattern Restricted Compositions

2019
*
Turkish Journal of Analysis and Number Theory
*

As

doi:10.12691/tjant-7-4-2
fatcat:jpmi7rw27rcgphl3qcowbiwnfu
*a*consequence*of*our results, we obtain all*of*the*distinct*distributions for the capacity statistic on avoidance classes*of*compositions corresponding to 3-*letter**patterns*having*at**most**two**distinct*... The total number*of*water cells*in*the bargraph representation*of**a**k*-*ary**word*then gives what is referred to as the capacity*of*w . ... τ has*length**three*and*at**most**two**distinct**letters*. ...##
###
Pattern statistics in faro words and permutations
[article]

2021
*
arXiv
*
pre-print

We study the distribution and the popularity

arXiv:2010.06270v4
fatcat:upz5rl3lcfandpczh4qvayadrm
*of*some*patterns**in**k*-*ary*faro*words*, i.e.*words*over the alphabet {1, 2, ...,*k*} obtained by interlacing the*letters**of**two*nondecreasing*words**of**lengths*... Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity*of**patterns**of**length**at**most**three*. ... p*of**length**at**most**three**in*faro permutations. ...##
###
Counting Words by Number of Occurrences of Some Patterns

2011
*
Electronic Journal of Combinatorics
*

We give asymptotic expressions for the number

doi:10.37236/630
fatcat:vojmkkznl5d53ccqm5cuuy2gr4
*of**words*containing*a*given number*of**occurrences**of**a**pattern*for*two*families*of**patterns**with**two*parameters each. ... One is the family*of*classical*patterns**in*the form $22\cdots 212 \cdots 22$ and the other is*a*family*of*partially ordered*patterns*. ... We begin by deriving*a*recursion for f r (n,*k*). We comment that this extends the work*in*[5] , that deals*with*avoidance*of*classical*patterns**with**at**most**two**distinct**letters*. ...##
###
Enumerative properties of restricted words and compositions
[article]

2019
*
arXiv
*
pre-print

For some problems our results extend to the case

arXiv:1811.10461v3
fatcat:4eruabslejfx5bjkzqbap63iei
*of**a*positive number*of*subword*pattern**occurrences*(instead*of*zero for*pattern*avoidance) or convergence*in*distribution*of*the normalized number*of**occurrences*...*In*this document we achieve exact and asymptotic enumeration*of**words*, compositions over*a*finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence*pattern*... Specifically, exact results for the avoidance*of*various subsequence*patterns**with**at**most*2*distinct**letters*were found*in*[17] . ...##
###
Squares in partial words

2014
*
Theoretical Computer Science
*

We also give

doi:10.1016/j.tcs.2014.02.023
fatcat:dzo4jr77abf2jew6palcetuhqm
*a*2k h tight bound on the number*of*rightmost*occurrences**of*squares per position*in**a**k*-*ary*partial*word**with*h holes. ...*of*square*occurrences**in**a*binary partial*word**with*h holes over its*length*n is 103/187, provided the limit*of*h/n is 0. ... We also thank Michelle Bodnar for her help*in*clarifying ideas*in*this paper. ...##
###
Segmental partially ordered generalized patterns

2005
*
Theoretical Computer Science
*

Mansour,

doi:10.1016/j.tcs.2005.09.036
fatcat:6eu3xguxe5f73evcv5h6xl4f2m
*Words*restricted by*patterns**with**at**most*2*distinct**letters*, Electron. J. Combin. 9 (2) (2002) #R3] for*words*. We deal*with*segmental POGPs (SPOGPs). ... The entire distribution*of*the maximum number*of*non-overlapping*occurrences**of**a*generalized*pattern**with*no dashes*in*permutations or*words*studied*in*[S. ... Acknowledgments The author is grateful to Richard Ehrenborg, Carl Lee, Margaret Readdy, and Peter Perry for their support during the author's stay*in*the Mathematics Department*at*the University*of*Kentucky ...##
###
Pattern Avoidance in Ternary Trees
[article]

2011
*
arXiv
*
pre-print

This paper considers the enumeration

arXiv:1110.2225v2
fatcat:xggk4hlmhjhuhjkl7ta72u63fi
*of*ternary trees (i.e. rooted ordered trees*in*which each vertex has 0 or 3 children) avoiding*a*contiguous ternary tree*pattern*. ... Finally, we compare our bijections to known "replacement rules" for binary trees and generalize these bijections to*a*larger class*of*trees. ... The authors thank Eric Rowland for*a*number*of*presentation suggestions and for support*with*generating the many tree graphics required for this paper. ...##
###
Counting words with Laguerre series
[article]

2013
*
arXiv
*
pre-print

We use this technique to find the generating function for the number

arXiv:1306.6232v1
fatcat:cficbxzwyngb7lp5jyoutbkd3e
*of**k*-*ary**words*avoiding any vincular*pattern*that has only ones, as well as*words*cyclically avoiding vincular*patterns**with*only ones ... We develop*a*method for*counting**words*subject to various restrictions by finding*a*combinatorial interpretation for*a*product*of*weighted sums*of*Laguerre polynomials*with*parameter \alpha = -1. ... For example,*a**word*contains 111-11 if it has five appearances*of*the same*letter**in*the*word*so that the first*three*and the last*two*are consecutive. ...##
###
FM-index for dummies
[article]

2015
*
arXiv
*
pre-print

After the first wave

arXiv:1506.04896v2
fatcat:yhdz2374ozakpgw4tw2vk4qssq
*of*interest*in*its theoretical developments, we can observe*a*surge*of*interest*in*practical FM-index variants*in*the last few years. ... The FM-index is*a*celebrated compressed data structure for full-text*pattern*searching. ... Acknowledgement We thank Simon Gog for helping us*in*using the sdsl-lite library and Shaun D. Jackman for an important remark concerning the ABySS de novo genome assembler. ...##
###
Packing a binary pattern in compositions

2011
*
Journal of Combinatorics
*

We are able to find the packing density for some classes

doi:10.4310/joc.2011.v2.n1.a5
fatcat:atse52ifmvgslhmmqyhfga655i
*of*subsequence and generalized*patterns*and all the*three**letter*binary*patterns*. ...*In*this article we generalize packing density problems from permutations and*words*to compositions. ... Acknowledgements The authors wish to thank an anonymous referee for substantial contributions on the proofs*of*Lemma 3.3 and Proposition 3.9. ...##
###
Counting occurrences of some subword patterns

2003
*
Discrete Mathematics & Theoretical Computer Science
*

*In*particular, we find generating functions for the number

*of*strings containing

*a*specified number

*of*

*occurrences*

*of*

*a*given 3-

*letter*subword

*pattern*. ... International audience We find generating functions the number

*of*strings (

*words*) containing

*a*specified number

*of*

*occurrences*

*of*certain types

*of*order-isomorphic classes

*of*substrings called subword ... The final version

*of*this paper was written while the second author (T.M.) was visiting University

*of*Haifa, Israel

*in*January 2003. ...

##
###
Toward the Combinatorial Limit Theory of Free Words
[article]

2015
*
arXiv
*
pre-print

*Word*V is

*a*factor

*of*

*word*W provided V occurs as consecutive

*letters*within W. W is an instance

*of*V provided there exists

*a*nonerasing monoid homomorphsism ϕ

*with*ϕ(V) = W. ... Free

*words*are elements

*of*

*a*free monoid, generated over an alphabet via the binary operation

*of*concatenation. Casually speaking,

*a*free

*word*is

*a*finite string

*of*

*letters*. ... On

*a*fixed q-

*letter*alphabet, there are fewer than q f(n,q)+1

*words*

*of*

*length*

*at*

*most*f(n, q).

*In*particular, there are fewer than q f(n,q)+1 Z n -instances

*of*

*length*

*at*

*most*f(n, q). ...

##
###
Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081)

2011
*
Dagstuhl Reports
*

Scientists working

doi:10.4230/dagrep.1.2.47
dblp:journals/dagstuhl-reports/CrochemoreKMN11
fatcat:s3v55bqr4fbdfo5kztirfoljqm
*in*the fields*of*Combinatorics on*Words*, Computational Biology, Stringology, Natural Computing, and Machine Learning were invited to consider the seminar's topic from*a*wide range*of*... Combinatorial problems about strings lie*at*the core*of*such algorithmic questions. ... Any*word*,*in*which some prefix coincides*with*some suffix, is*a*fractional power. ...##
###
On Searching Zimin Patterns
[article]

2014
*
arXiv
*
pre-print

Finally, we give some bounds on the function f(n,

arXiv:1409.8235v1
fatcat:n6tjyq4gzfcrfjwhpohhyc3mtu
*k*) such that every*k*-*ary**word**of**length**at*least f(n,*k*) has*a*factor that matches the rank n Zimin*pattern*. ... Zimin type*of**a**word*x is introduced here as the maximum rank*of**a*Zimin*pattern*matching x. We show how to compute Zimin type*of**a**word*on-line*in*linear time. ... The proof is based on*two*lemmas. (2) The*length**of**a*Fibonacci factor*of*period Φ*k*is*at**most*Φ*k*+1 + 2Φ*k*− 2. Lemma 3.7. ...##
###
Searching for Zimin patterns

2015
*
Theoretical Computer Science
*

Finally, we give some bounds on the function f (n,

doi:10.1016/j.tcs.2015.01.004
fatcat:zy3ngm3kqbhxtdqqqdhxerzpq4
*k*) such that every*k*-*ary**word**of**length**at*least f (n,*k*) has*a*factor that matches the rank n Zimin*pattern*. ... Zimin type*of**a**word*x is introduced here as the maximum rank*of**a*Zimin*pattern*matching x. We show how to compute Zimin type*of**a**word*on-line*in*linear time. ... The proof is based on*two*lemmas. (2) The*length**of**a*Fibonacci factor*of*period Φ*k*is*at**most*Φ*k*+1 + 2Φ*k*− 2. Lemma 3.7. ...
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