• Mark Shattuck

      Articles written in Proceedings – Mathematical Sciences

    • Enumerating Set Partitions According to the Number of Descents of Size 𝑑 or more

      Toufik Mansour Mark Shattuck Chunwei Song

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      Let $P(n,k)$ denote the set of partitions of $\{1,2,\ldots,n\}$ having exactly 𝑘 blocks. In this paper, we find the generating function which counts the members of $P(n,k)$ according to the number of descents of size 𝑑 or more, where $d\geq 1$ is fixed. An explicit expression in terms of Stirling numbers of the second kind may be given for the total number of such descents in all the members of $P(n,k)$. We also compute the generating function for the statistics recording the number of ascents of size 𝑑 or more and show that it has the same distribution on $P(n,k)$ as the prior statistics for descents when $d\geq 2$, by both algebraic and combinatorial arguments.

    • A Statistic on 𝑛-Color Compositions and Related Sequences

      Toufik Mansour Mark Shattuck

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      A composition of a positive integer in which a part of size 𝑛 may be assigned one of 𝑛 colors is called an 𝑛-color composition. Let $a_m$ denote the number of 𝑛-color compositions of the integer 𝑚. It is known that $a_m = F_{2m}$ for all $m \geq 1$, where $F_m$ denotes the Fibonacci number defined by $F_m = F_{m-1}+F_{m-2}$ if $m\geq 2$, with $F_0=0$ and $F_1=1$. A statistic is studied on the set of 𝑛-color compositions of 𝑚 thus providing a polynomial generalization of the sequence $F_{2m}$. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of 𝑛-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide 𝑞-generalizations of relations between $a_m$ and the number of self-inverse 𝑛-compositions of $2m+1$ or $2m$. Finally, we consider a more general recurrence than that satisfied by the numbers $a_m$ and note some particular cases.

    • Generalized $r$-Lah numbers


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      In this paper, we consider a two-parameter polynomial generalization, denoted by ${\mathcal G}_{a,b}(n, k; r)$, of the $r$-Lah numbers which reduces to these recently introduced numbers when $a = b = 1$. We present several identities for ${\mathcal G}_{a,b}(n, k; r)$ that generalize earlier identities given for the $r$-Lah and $r$-Stirling numbers. We also provide combinatorial proofs of some earlier identities involving the $r$-Lah numbers by defining appropriate sign-changing involutions. Generalizing these arguments yields orthogonality-type relations that are satisfied by ${\mathcal G}_{a,b}(n, k; r)$.

    • Counting rises and levels in $r$-color compositions


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      An $r$-color composition of a positive integer $n$ is a sequence of positiveintegers, called parts, summing to n in which each part of size $r$ is assigned one of $r$ possible colors. In this paper, we address the problem of counting the $r$-color compositions having a prescribed number of rises. Formulas for the relevant generating functions are computed which count the compositions in question according to a certain statistic. Furthermore, we find explicit formulas for the total number of rises within all of the $r$-color compositions of $n$ having a fixed number of parts. A similar treatment is given for the problem of counting the number of levels and a further generalization in terms of rises of a particular type is discussed.

    • Determinant formulas of some Toeplitz–Hessenberg matrices with Catalan entries


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      In this paper, we consider determinants of some families of Toeplitz–Hessenberg matrices having various translates of the Catalan numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Catalan numbers and multinomial coefficients. Combinatorial proofs may be given for several of the identities that are obtained.

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