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Counting Water Cells in Pattern Restricted Compositions

Toufik Mansour, Mark Shattuck
2019 Turkish Journal of Analysis and Number Theory  
As 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.  ... 
doi:10.12691/tjant-7-4-2 fatcat:jpmi7rw27rcgphl3qcowbiwnfu

Pattern statistics in faro words and permutations [article]

Jean-Luc Baril, Alexander Burstein, Sergey Kirgizov
2021 arXiv   pre-print
We study the distribution and the popularity 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.  ... 
arXiv:2010.06270v4 fatcat:upz5rl3lcfandpczh4qvayadrm

Counting Words by Number of Occurrences of Some Patterns

Zhicheng Gao, Andrew MacFie, Daniel Panario
2011 Electronic Journal of Combinatorics  
We give asymptotic expressions for the number 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.  ... 
doi:10.37236/630 fatcat:vojmkkznl5d53ccqm5cuuy2gr4

Enumerative properties of restricted words and compositions [article]

Andrew MacFie
2019 arXiv   pre-print
For some problems our results extend to the case 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] .  ... 
arXiv:1811.10461v3 fatcat:4eruabslejfx5bjkzqbap63iei

Squares in partial words

F. Blanchet-Sadri, Yang Jiao, John M. Machacek, J.D. Quigley, Xufan Zhang
2014 Theoretical Computer Science  
We also give 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.  ... 
doi:10.1016/j.tcs.2014.02.023 fatcat:dzo4jr77abf2jew6palcetuhqm

Segmental partially ordered generalized patterns

Sergey Kitaev
2005 Theoretical Computer Science  
Mansour, 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  ... 
doi:10.1016/j.tcs.2005.09.036 fatcat:6eu3xguxe5f73evcv5h6xl4f2m

Pattern Avoidance in Ternary Trees [article]

Nathan Gabriel, Katherine Peske, Lara Pudwell, Samuel Tay
2011 arXiv   pre-print
This paper considers the enumeration 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.  ... 
arXiv:1110.2225v2 fatcat:xggk4hlmhjhuhjkl7ta72u63fi

Counting words with Laguerre series [article]

Jair Taylor
2013 arXiv   pre-print
We use this technique to find the generating function for the number 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.  ... 
arXiv:1306.6232v1 fatcat:cficbxzwyngb7lp5jyoutbkd3e

FM-index for dummies [article]

Szymon Grabowski and Marcin Raniszewski and Sebastian Deorowicz
2015 arXiv   pre-print
After the first wave 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.  ... 
arXiv:1506.04896v2 fatcat:yhdz2374ozakpgw4tw2vk4qssq

Packing a binary pattern in compositions

Ragnar Freij, Toufik Mansour
2011 Journal of Combinatorics  
We are able to find the packing density for some classes 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.  ... 
doi:10.4310/joc.2011.v2.n1.a5 fatcat:atse52ifmvgslhmmqyhfga655i

Counting occurrences of some subword patterns

Alexander Burstein, Toufik Mansour
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.  ... 
doi:10.46298/dmtcs.320 fatcat:imenj3ecmrgwvob7bmzffmqwbu

Toward the Combinatorial Limit Theory of Free Words [article]

Danny Rorabaugh
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).  ... 
arXiv:1509.04372v1 fatcat:r2xc4du7szhathdmb2ssyjhmji

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

Maxime Crochemore, Lila Kari, Mehryar Mohri, Dirk Nowotka, Marc Herbstritt
2011 Dagstuhl Reports  
Scientists working 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.  ... 
doi:10.4230/dagrep.1.2.47 dblp:journals/dagstuhl-reports/CrochemoreKMN11 fatcat:s3v55bqr4fbdfo5kztirfoljqm

On Searching Zimin Patterns [article]

Wojciech Rytter, Arseny M. Shur
2014 arXiv   pre-print
Finally, we give some bounds on the function f(n,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.  ... 
arXiv:1409.8235v1 fatcat:n6tjyq4gzfcrfjwhpohhyc3mtu

Searching for Zimin patterns

Wojciech Rytter, Arseny M. Shur
2015 Theoretical Computer Science  
Finally, we give some bounds on the function f (n, 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.  ... 
doi:10.1016/j.tcs.2015.01.004 fatcat:zy3ngm3kqbhxtdqqqdhxerzpq4
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