M. Shattuck

Publications146
h-index12
Citations601
Highly Influential Citations36
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Pattern avoidance in inversion sequences
Abstract A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 < ai < i for all z, which is called an inversion sequence. In analogy to the usual case… Expand
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Counting humps and peaks in generalized Motzkin paths
TLDR
We relate the total number of humps in all of the (k,a)-paths of order n to the number of super (k-a-paths, where a hump is defined to be a sequence of steps of the form UH^iD. Expand
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Some enumerative results related to ascent sequences
TLDR
An ascent sequence is a sequence of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Expand
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Some Wilf-equivalences for vincular patterns
We prove several Wilf-equivalences for vincular patterns of length 4, some of which generalize to infinite families of vincular patterns. We also present functional equations for the generating… Expand
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A general two-term recurrence and its solution
TLDR
We find a general explicit formula for all sequences satisfying a two-term recurrence of a certain kind. Expand
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Combinatorial proofs of some Simons-type binomial coefficient identities.
In this note, we present combinatorial proofs of some Moriarty-type binomial coe! cient identities using linear and circular domino arrangements.
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Combinatorial trigonometry with Chebyshev polynomials
We provide a combinatorial proof of the trigonometric identity cosðny Þ¼ TnðcosyÞ, where Tn is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric… Expand
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A $$q$$-analog of the hyperharmonic numbers
Recently, the $$q$$-analog of the harmonic numbers obtained by replacing each positive integer $$n$$ with $$n_q$$ has been shown to satisfy congruences which generalize Wolstenholme’s theorem. Here,… Expand
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Counting Peaks and Valleys in a Partition of a Set
A partitionof the set (n) = {1,2,...,n} is a collection {B1,B2,...,Bk} of nonempty disjoint subsets of (n) (called blocks) whose union equals (n). In this paper, we find an explicit formula for the… Expand
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On a New Family of Generalized Stirling and Bell Numbers
TLDR
A family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. Expand
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