S. Kitaev, T. Mansour
We introduce Peano words, which are words corresponding to finite approximations of the Peano space filling curve. We then find the number of occurrences of certain patterns in these words.
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A double occurrence word $w$ over a finite alphabet $Σ$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence words have been used for studying DNA recombination events. We develop formulas for counting and enumerating several elementary classes of double occurrence words such as palindromic, irreducible, and strongly-irreducible words.
We consider sigma-words, which are words used by Evdokimov in the construction of the sigma-sequence. We then find the number of occurrences of certain patterns and subwords in these words.
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 patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern.
We study the generating function for the number of involutions on $n$ letters containing exactly $r\gs0$ occurrences of 3412. It is shown that finding this function for a given $r$ amounts to a routine check of all involutions on $2r+1$ letters.
We study the generating function for the number of even (or odd) permutations on n letters containing exactly $r\gs0$ occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in $S_{2r}$.
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a m…
We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems -- particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension 2 posets is #P-hard; enumerating the linear extensions that occur in realizers of dimension 2 posets can be done in polynomial time, while for uncons…
Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type $(1,2)$ or $(2,1)$. For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. With respect to being equidistributed there are three different classes of patterns of type $(1,2)$ or $(2,1)$. We present a recursion for the number of perm…
Counting occurrences of a pattern in a binary word
本文提出字典序极值引用方法，给出二元词中出现次数的计算公式，并构造算法枚举包含给定词 p 的二元词及本原词，以研究 B_{n,p}(k)。
We apply ideas from the cluster method to q-count the permutations of a multiset according to the number of occurrences of certain generalized patterns, as defined by Babson and Steingrimsson. In particular, we consider those patterns with three letters and one internal dash, as well as permutation statistics composed of counting the number of occurrences of multisets of such patterns. Counting is done via recurrences which simplify in the case of permutations. A collection of Maple procedures implementing these recurrences accompanies the article.
The word inference problem is to determine languages such that the information on the number of occurrences of those subwords in the language can uniquely identify a word. A considerable amount of work has been done on this problem, but the same cannot be said for circular words despite growing interests on the latter due to their applications -- for example, in splicing systems. Meanwhile, Parikh matrices are useful tools and well established in the study of subword occurrences. In this work, we propose two ways of counting subword occurrences in circular words. We then extend the idea of Par…
We count the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. More precisely, we consider the sequence $(S_b(n))_{n\ge 0}$ counting the number of positive entries on each row of a generalization of the Pascal triangle to binomial coefficients of base-$b$ expansions. By using a convenient tree structure, we provide recurrence relations for $(S_b(n))_{n\ge 0}$ leading to the $b$-regularity of the latter sequence. Then we deduce the asymptotics of the summatory function of the sequence $(S_b(n))_{n\ge 0}$.
Counting occurrences of patterns in permutations
本文提出一种多重集合元素计数新方法，并将其用于研究排列中模式出现次数，扩展长度3模式结果，系统分析长度4模式并猜想七个Wilf类的渐近行为。
For a given language $L$, we study the languages $X$ such that for all distinct words $u, v \in L$, there exists a word $x \in X$ that appears a different number of times as a factor in $u$ and in $v$. In particular, we are interested in the following question: For which languages $L$ does there exist a finite language $X$ satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.
We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method.
The Goulden-Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of prescribed consecutive patterns. In this paper, we lift the cluster method for permutations to the Malvenuto-Reutenauer algebra. Upon applying standard homomorphisms, our result specializes to both the cluster method for permutations as well as a q-analogue which keeps track of the inversion number statistic. We construct additional homomorphisms using the theory of shuffle-compatibility, leading to furth…
In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern $σ$ beginning with 1, the bivariate generating function counting occurrences of $σ$ in circular permutations can be obtained from the generating function counting occurrences of $σ$ in (linear) permutations. This includes all the patterns for which the latter generating function is known.
Given a set $Π$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(Π)$, the set of permutations of length at most $n$ avoiding the patterns in $Π$, in time $O(|S_{\le n - 1}(Π)| \cdot k + |S_{n}(Π)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $Π$ in each permutation in $S_n$. Surprisingly, when $|Π| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutat…
Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended and generalized in several ways. In this paper, we shall fix a set partition of the natural numbers $N$, $(N_1, ..., N_t)$, and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in $N_i$ over the set of words over the alphabet $[k]$. In particular, we refine and g…
Counting $r\times s$ rectangles in (Catalan) words
本文提出字的矩形容量统计量以计数条形图中的 $r\ imes s$ 矩形，并推导普通字与 Catalan 字的分布及总数生成函数。
Enumeration of Factor Occurrences in $k$-Bonacci Words over an Infinite Alphabet
本文研究无限字母集上的k-拜占庭词，通过有限迭代次数分析因子出现统计，推导出k≥3时任意数字在第n次迭代中出现次数的生成函数，并描述了所有长度为2的因子出现次数的生成函数。
Classical pattern distributions in $\mathcal{S}_{n}(132)$ and $\mathcal{S}_{n}(123)$
本研究针对132-避免与123-避免的置换集合，通过推导生成函数的显式递推关系，刻画其中经典模式的出现计数分布，补充了对称群中模式分布研究的相关结论。
Counting $r\times s$ rectangles in nondecreasing and Smirnov words
本文研究了非降字与 Smirnov 字中 $r\ imes s$ 矩形容量的分布，求得了其二元生成函数及总数生成函数，补充了关于一般字和 Catalan 字的近期研究。
The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions for counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths---both with height bounded and unbounded---by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued f…
An avoidance pattern where the letters within an occurrence of which are required to be adjacent is referred to as a subword. In this paper, we enumerate members of the set NC_n of non-crossing partitions of length n according to the number of occurrences of several infinite families of subword patterns each containing repeated letters. As a consequence of our results, we obtain explicit generating function formulas counting the members of NC_n for n >= 0 according to all subword patterns of length three containing a repeated letter. Further, simple expressions are deduced for the total number…
Statistics on clusters and $r$-Stirling permutations
本文推广Goulden-Jackson簇方法，研究了$2134\cdots m$-簇按逆下降数等统计量的计数，推导出排列计数公式，并发现其与$r$-Stirling排列同数及联合同分布性质。
Block-counting sequences are not purely morphic
本文研究m进制展开中有限字出现次数模p构成的序列，提出快速生成算法，证明素数m下多数序列为m-一致但非纯形态，且对应幂级数在F_m(t)上为m次代数级数。
On the Partial Sum of Subword-Counting Sequences
本文研究了二进制展开中子词出现次数的部分和 $\sum_{n=0}^N(-1)^{s_w(n)}$ 的行为，并刻画了满足该部分和为 $O(N^{1-ε})$ 的几类词 $w$。
Pattern containment in random permutations
本文研究随机排列中计数模式出现次数的统计量，通过将模式计数统计量的均值表示为不可约特征标的线性组合，给出了计算经典模式和 vincular 模式期望值的完整方法。
One of the most challenging problems in enumerative combinatorics is to count Wilf classes, where you are given a pattern, or set of patterns, and you are asked to find a "formula", or at least an efficient algorithm, that inputs a positive integer n and outputs the number of permutations avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a prescribed, r, say, occurrences of a given pattern. They gave an ingenious method to generate Functional Equations, alas, with an unbounded number of "catalytic variables", but then described a …
We consider the problem of counting the occurrences of patterns of the form $xy-z$ within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on $\mathcal{S}_n$ for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a unified approach to obtain explicit formulas. By these recurrences, we are able to determine simple closed form expressions for the number of permutations that, when flattened, avoid one of these patterns as well as expressions for the average number of occurrences. In particular, we find that the avera…
Counting Permutation Patterns with Multidimensional Trees
本文提出 pattern-trees 及其求值框架，解决排列中所有长度为 k 的模式计数问题，给出 k=6、7 的近二次时间算法和 k=5 的近 $n^{7/4}$ 时间算法，并指出 k=8 的维数障碍。
Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length $n$ contains a given pattern of length $k$. In this work we give two new algorithms for this well-studied problem, one whose running time is $n^{k/4 + o(k)}$, and a polynomial-space algorithm whose running time is the better of $O(1.6181^n)$ and $O(n^{k/2 + 1})$. These results improve the earlier best bo…
A sample of n generic points in the xy-plane defines a permutation that relates their ranks along the two axes. Every subset of k points similarly defines a pattern, which occurs in that permutation. The number of occurrences of small patterns in a large permutation arises in many areas, including nonparametric statistics. It is therefore desirable to count them more efficiently than the straightforward ~O(n^k) time algorithm. This work proposes new algorithms for counting patterns. We show that all patterns of order 2 and 3, as well as eight patterns of order 4, can be counted in nearly linea…
We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function. We also complete the classification of consecutive …
In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding involutions and Dyck word prefixes of same length. Extending this bijection to bilateral words allows to determine more parameters; in particular, we consider the number of inversions and rises of the involutions onto the words. This is the starting point for considering two different directions: even/odd involutions and statistics of some generalized patterns. T…
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