Mark Shattuck
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View article: Counting adjacencies with difference at most one in -ary words
Counting adjacencies with difference at most one in -ary words Open
In this paper, we study the distribution on \([k]^n\) for the parameter recording the number of indices \(i \in [n-1]\) within a word \(w=w_1\cdots w_n\) such that \(|w_{i+1}-w_i|\ \leq 1\) and compute the corresponding (bivariate) generat…
View article: On ascent sequences avoiding 021 and a pattern of length four
On ascent sequences avoiding 021 and a pattern of length four Open
Ascent sequences of length $n$ avoiding the pattern $021$ are enumerated by the $n$-th Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$. In this paper, we extend this result and enumerate ascent sequences avoiding $\{021,τ\}$, where $τ$ is …
View article: Toeplitz–Hessenberg determinant formulas for the sequence \(F_n-1\)
Toeplitz–Hessenberg determinant formulas for the sequence \(F_n-1\) Open
Let Fn denote the n-th Fibonacci number defined by Fn = Fn − 1 + Fn − 2 if n ≥ 2, with F0 = 0 and F1 = 1. In this paper, we find determinant identities for several Toeplitz–Hessenberg matrices whose nonzero entries are derived from the seq…
View article: Further results for the capacity statistic distribution on compositions of 1s and 2s
Further results for the capacity statistic distribution on compositions of 1s and 2s Open
In this paper, we study additional aspects of the capacity distribution on the set ℬn of compositions of n consisting of 1’s and 2’s extending recent results of Hopkins and Tangboonduangjit. Among our results are further recurrences for th…
View article: Counting k-ary words by number of adjacency differences of a prescribed size
Counting k-ary words by number of adjacency differences of a prescribed size Open
Recently, the general problem of enumerating permutations $π=π_1\cdots π_n$ such that $π_{i+r}-π_i \neq s$ for all $1\leq i\leq n-r$, where $r$ and $s$ are fixed, was considered by Spahn and Zeilberger. In this paper, we consider an analog…
View article: Enumeration of consecutive patterns in flattened Catalan words
Enumeration of consecutive patterns in flattened Catalan words Open
A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has…
View article: Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's
Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's Open
In this paper, we study additional aspects of the capacity distribution on the set $\mathcal{B}_n$ of compositions of $n$ consisting of $1$'s and $2$'s. Among our results are further recurrences for this distribution as well as formulas fo…
View article: Enumeration of non-crossing partitions according to subwords with repeated letters
Enumeration of non-crossing partitions according to subwords with repeated letters Open
In this paper, we enumerate members of the set NCn 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 res…
View article: Enumeration of inversion sequences according to the outer and inner perimeter
Enumeration of inversion sequences according to the outer and inner perimeter Open
The integer sequence π = π 1 ‧‧‧ π n is said to be an inversion sequence if 0 ≤ π i ≤ i – 1 for all i . Let ℐ n denote the set of inversion sequences of length n , represented using positive instead of non-negative integers. We consider he…
View article: Avoidance of vincular patterns by Catalan words
Avoidance of vincular patterns by Catalan words Open
Let $\mathcal{C}_n$ denote the set of words $w=w_1\cdots w_n$ on the alphabet of positive integers satisfying $w_{i+1}\leq w_i+1$ for $1 \leq i \leq n-1$ with $w_1=1$. The members of $\mathcal{C}_n$ are known as Catalan words and are enume…
View article: Counting subword patterns in Catalan words
Counting subword patterns in Catalan words Open
Let An denote the set of integral sequences w = w1⋯wn such that wi + 1 ≤ wi + 1 for 1 ≤ i ≤ n − 1, with w1 = 1. In this paper, we enumerate the members of An, known as Catalan words, according to the number of occurrences of any subword pa…
View article: Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries
Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries Open
Let u n = u n ( k ) denote the generalized Leonardo number defined recursively by u n = u n− 1 + u n− 2 + k for n ≥ 2, where u 0 = u 1 = 1. Terms of the sequence u n (1) are referred to simply as Leonardo numbers. In this paper, we find ex…
View article: Hessenberg-Toeplitz matrix determinants with Schröder and Fine number entries
Hessenberg-Toeplitz matrix determinants with Schröder and Fine number entries Open
In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schröder and Fine number sequences. Algebraic proofs of these results can be given which make us…
View article: Determinant identities for the Catalan, Motzkin and Schröder numbers
Determinant identities for the Catalan, Motzkin and Schröder numbers Open
In this paper, we find formulas for the determinants of several Hessenberg matrices whose nonzero entries are derived from the Catalan, Motzkin and Schröder number sequences. By a generalization of Trudi's formula, we obtain equivalent mul…
View article: Hessenberg-Toeplitz Matrix Determinants with Schroder and Fine Number Entries
Hessenberg-Toeplitz Matrix Determinants with Schroder and Fine Number Entries Open
In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schroder and Fine number sequences. Algebraic proofs of these results can be given which make us…
View article: Enumeration of non-crossing partitions according to subwords with repeated letters
Enumeration of non-crossing partitions according to subwords with repeated letters Open
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 numbe…
View article: Combinatorial proofs of identities for the generalized Leonardo numbers
Combinatorial proofs of identities for the generalized Leonardo numbers Open
In this paper, we provide combinatorial proofs of several prior identities satisfied by the recently introduced generalized Leonardo numbers, denoted by \mathcal{L}_{k,n}, as well as derive some new formulas. To do so, we interpret \mathca…
View article: Counting occurrences of subword patterns in non-crossing partitions
Counting occurrences of subword patterns in non-crossing partitions Open
A permutation pattern in which all letters within an occurrence are required to be adjacent is known as a subword. In this paper, we consider the distribution of several infinite families of subword patterns on the set of non-crossing part…
View article: A Further Look at the Bayes Blind Spot
A Further Look at the Bayes Blind Spot Open
Gyenis and Redei have demonstrated that any prior p on a finite algebra, however chosen, severely restricts the set of posteriors accessible from p by Jeffrey conditioning on a nontrivial partition. Their demonstration involves showing tha…
View article: Statistics on bargraphs of inversion sequences of permutations
Statistics on bargraphs of inversion sequences of permutations Open
We consider the joint distribution of the area and perimeter statistics on the set I_n of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived…
View article: Identities relating six members of the Fibonacci family of sequences
Identities relating six members of the Fibonacci family of sequences Open
In this paper, we prove several identities each relating a sum of products of three terms coming from different members of the Fibonacci family of sequences with a comparable sum whose terms come from three other sequences. These identitie…
View article: Further enumeration results concerning a recent equivalence of restricted inversion sequences
Further enumeration results concerning a recent equivalence of restricted inversion sequences Open
Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inv…
View article: Some New Families of Compositions Based on Big Part Restrictions
Some New Families of Compositions Based on Big Part Restrictions Open
In this paper, we enumerate four new families of compositions whose members satisfy certain conditions on the sizes of the big (i.e., > 1) parts and/or lengths of the 1-strings.In particular, we consider various classes of compositions who…
View article: Counting subword patterns in permutations arising as flattened partitions of sets
Counting subword patterns in permutations arising as flattened partitions of sets Open
We consider various statistics on the set Fn consisting of the distinct permutations of length n+1 that arise as a flattening of some partition of the same size. In particular, we enumerate members of Fn according to the number of occurren…
View article: Fibonacci–Lucas–Pell–Jacobsthal relations
Fibonacci–Lucas–Pell–Jacobsthal relations Open
In this paper, we prove several identities involving linear combinations of convolutions of the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes of second-order linearly recurrent sequences wit…
View article: A formula relating Bell polynomials and Stirling numbers of the first kind
A formula relating Bell polynomials and Stirling numbers of the first kind Open
In this paper, we prove a general convolution formula involving the Bell polynomials and the Stirling numbers of the first kind.Our proof of the formula is algebraic and establishes an equivalent identity involving the associated exponenti…