Related papers: Integer Sequences: Irregular Arrays and Intra-Bloc…
We propose several procedures for creating new families of integer sequences based on the method of Cantor diagonalization. Then we modify and generalize this method. The paper includes explicit formulas for most proposed families of…
A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…
Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of…
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…
Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural…
In this paper, we give recurrence relations and identities for some integer sequences related to Ward numbers such as Ward-Lah numbers, varied Ward numbers and binomial Ward numbers. Most of the sequences are entered in the On-Line…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic…
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…
This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the…
Compression of integer sets and sequences has been extensively studied for settings where elements follow a uniform probability distribution. In addition, methods exist that exploit clustering of elements in order to achieve higher…
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the…
There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide…
The Hankel transform of an integer sequence is a much studied and much applied mathematical operation. In this note, we extend the notion in a natural way to sequences of $d$ integer sequences. We explore links to generalized continued…
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of…
We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and…
We use the concept of the half of a lower-triangular matrix to define a transformation on integer sequences. We explore the properties of this transformation, including in some cases a study of the Hankel transform of the transformed…